12 Commuting Parent Hamiltonians and Spectral Gaps
This chapter studies commuting parent Hamiltonians and spectral gaps for matrix product states. For normal tensors, the first sections compare commutation of neighboring parent terms with renormalization fixed points under the stated canonical and ground-space hypotheses. Decorrelation is expressed through commuting support projectors and their idempotent product. For frustration-free sums of local projections, spectral-gap estimates are established via the anticommutator martingale criterion and cyclic norm-compression bounds, yielding uniform lower bounds on the gap above the ground space.
12.1 Commuting Local Terms for Parent Hamiltonians
This section concerns the commutation equation for local terms in a periodic chain, relating renormalization fixed points to tensors with commuting nearest-neighbor terms, \(h_i(A,2)h_j(A,2)=h_j(A,2)h_i(A,2)\) [ 13 , Definition 3.9 and Theorem 3.10 ] . The reverse implication from commuting local terms to a renormalization fixed point is proved for normal left-canonical BNT representatives assuming the all-chain ground-space condition (zero energy and kernel spanned by the periodic MPS vectors for every \(N{\gt}2\)).
For every type \(\alpha \), evaluation at the three coordinates gives the canonical equivalence
Its forward map is
On a three-site space \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\), let \(Q_{AX}\) and \(Q_{XB}\) be two-site projectors. Their local actions on the three-site space are the operators
The algebraic part of the overlapping nearest-neighbor commutation condition is
after these two local actions are viewed on the common three-site space. This is the local-support part of the parent commuting Hamiltonian condition in [ 13 , Definition D.2 ] . The cyclic windows beginning at \(i\) and \(i+1\) realize the two supports \(AX\) and \(XB\), and hence overlap at the middle site.
The source local condition also fixes a three-site subspace \(K_{AXB}\) and imposes the kernel-intersection equation \(K_{AXB}=\ker ((Q_{AX})_{AX})\cap \ker ((Q_{XB})_{XB})\). Forgetting this intersection equation gives the algebraic overlapping two-site commutation condition displayed above.
The two lift maps also distribute over complementary projectors:
Consequently, if the projectors \(Q_{AX}\) and \(Q_{XB}\) commute on the three-site space, then their complementary projectors \(P_{AX}=1-Q_{AX}\) and \(P_{XB}=1-Q_{XB}\) commute after the same lifts. This is the elementary algebraic passage used when one moves between the \(Q\)-projectors of [ 13 , Appendix D.2 ] and the complementary ground-space projectors \(P\). Since the original two projectors are idempotent, the complementary projectors are idempotent as well; hence the pair \(P_{AX},P_{XB}\) satisfies the same idempotent-and-commutation condition on the three-site space.
Let \(\mathcal{H}=\mathbb {C}^d\), and let \(Q\) be an endomorphism of \(\mathcal{H}\otimes \mathcal{H}\). Under the canonical identifications of two- and three-site configurations with pairs and right-associated triples, the matrices of its two adjacent actions are
Equivalently, for all \(a,x,b,a',x',b'\in \{ 0,\ldots ,d{-}1\} \),
This is the homogeneous three-single-site specialization of the factor actions on \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\) in [ 13 , Definition D.2 ] ; here \(\mathcal{H}_A=\mathcal{H}_X=\mathcal{H}_B=\mathcal{H}\). No assertion is made for unequal factor spaces.
On the first pair, the final coordinate is unchanged, which contributes \(\delta _{b,b'}\) to the matrix coefficient. On the final pair, the first coordinate is unchanged, which contributes \(\delta _{a,a'}\).
Put \(q(A)=1-P_{G_2(A)}\) for the canonical two-site parent interaction. On the three-site space \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\), with sites \(0,1,2\) labelled \(A,X,B\), the two adjacent translated length-two parent interactions are the two local actions
This identifies the two translated parent interactions with the local actions in [ 13 , Definition D.2 ] .
Let \(\sigma =(a,x,b)\) be a three-site configuration. For the first translated term, the length-two window and the corresponding replacement satisfy
For the second translated term,
Substituting these two pairs of identities into the definition of the translated parent term gives the two displayed equations.
Suppose the canonical two-site parent interaction \(q(A)=1-P_{G_2(A)}\) satisfies the overlapping \(AX/XB\) commutation condition of Definition 12.1.2. Then, on the three-site chain, \(h^{(3)}_0(A,2)h^{(3)}_1(A,2) = h^{(3)}_1(A,2)h^{(3)}_0(A,2)\). This only identifies the first two translated local terms with the lifted \(AX\) and \(XB\) actions; the construction of the [ 13 , Appendix D.2 ] projectors associated with the [ 13 , Appendix B ] basic-vector form is a separate step.
By Theorem 12.1.4, the two translated local terms are \(q(A)_{AX}\) and \(q(A)_{XB}\). The assumed overlapping support condition says exactly that these two lifted endomorphisms commute.
Let \(Q_{AX}\) and \(Q_{XB}\) be two-site projectors whose lifted \(AX\) and \(XB\) actions commute. Suppose that \(q(A)=Q_{AX}\) and \(q(A)=Q_{XB}\). Then the first two translated length-two local terms on the three-site chain commute.
The lift identities identify the translated terms with the lifted actions of the two projectors:
The overlapping-support hypothesis is the equation \((Q_{AX})_{AX}(Q_{XB})_{XB} = (Q_{XB})_{XB}(Q_{AX})_{AX}\). Substituting the two lifted identities gives the desired commutation of \(h^{(3)}_0(A,2)\) and \(h^{(3)}_1(A,2)\).
Suppose the projectors \(Q_{AX}\) and \(Q_{XB}\) satisfy the [ 13 , Definition D.2 ] local parent-commuting condition on \(A X B\), and suppose that the canonical two-site parent interaction is identified, as a two-site operator, with the two source projectors: \(q(A)=Q_{AX}\) and \(q(A)=Q_{XB}\). After lifting these two equalities to the \(AX\) and \(XB\) faces, the first two translated length-two local terms on the three-site chain commute. Thus, \(h^{(3)}_0(A,2)h^{(3)}_1(A,2) = h^{(3)}_1(A,2)h^{(3)}_0(A,2)\).
Suppose the Appendix D.2 projectors \(Q_{AX}\) and \(Q_{XB}\) satisfy the local parent-commuting condition on \(A X B\), and suppose that the canonical two-site parent interaction is their complement on both faces: \(q(A)=1-Q_{AX}=1-Q_{XB}\). Then the first two translated length-two local terms on the three-site chain commute, \(h^{(3)}_0(A,2)h^{(3)}_1(A,2) = h^{(3)}_1(A,2)h^{(3)}_0(A,2)\).
The lift identities of Theorem 12.1.4 identify the translated terms with \((1-Q_{AX})_{AX}\) and \((1-Q_{XB})_{XB}\). The Appendix D.2 condition implies the overlapping two-site commutation condition after forgetting the kernel-intersection equation. Definition 12.1.2 then gives commutation of the complementary lifted projectors, hence the displayed commutation relation.
The parent Hamiltonian with block length \(L\) on \(N\) sites has commuting local terms if \(h_i h_j=h_j h_i\) for all \(i,j\in \{ 0,\ldots ,N{-}1\} \). This records the translated commutation equation \([\tau _j(P_L),P_L]=0\) from [ 13 , Definition 3.9 ] ; the same source definition also requires the ground space to be spanned, for \(N{\gt}L\), by the periodic MPS vectors of the BNT components.
The nearest-neighbor commuting condition is the length-\(2\) specialization of the preceding commutation equation. For every \(i,j\in \{ 0,\ldots ,N{-}1\} \), one has \(h_i h_j=h_j h_i\). The source definition [ 13 , Definition 3.9 ] also requires the corresponding parent Hamiltonian to have \(L=2\) and to have the prescribed ground-space spanning property for \(N{\gt}2\).
Let \(L\le N\). Suppose that \(h_i h_j=h_j h_i\) whenever the length-\(L\) cyclic supports of the two local terms meet. Then the parent Hamiltonian has commuting local terms: \(h_i h_j=h_j h_i\) for all \(i,j\). In the case \(L=2\), this gives the nearest-neighbor commuting condition.
If the two cyclic supports meet, the hypothesis gives the commutation relation. If they are disjoint, the corresponding local terms commute by locality.
For \(i,j\in \mathbb {Z}/N\mathbb {Z}\), if the two length-two cyclic supports starting at \(i\) and \(j\) overlap, then
Expanding each length-two cyclic support gives shifts \(r_i,r_j\in \{ 0,1\} \) and an equality of sites \(i+r_i\equiv j+r_j\pmod N\). The four possible pairs \((r_i,r_j)\) give exactly the three alternatives in (3).
Let \(N\ge 2\). Suppose that, for every \(i\in \mathbb {Z}/N\mathbb {Z}\), the translated length-two local term commutes with its cyclic right neighbor:
Then all translated length-two local terms commute.
This condition records the commutation equation and the zero-energy equation for the nearest-neighbor local terms:
This is the zero-energy part of the condition appearing in [ 13 , Theorem 3.10(iii) ] : the MPS vector has zero energy for the translated two-site local terms, and those terms commute. The missing source assertion is that the whole parent-Hamiltonian ground space is spanned by the corresponding periodic MPS vectors from the BNT components, as recorded in Definition 12.1.15.
Let \(B\) be a canonical-form tensor with BNT components \(A_1,\ldots ,A_g\). The source parent-Hamiltonian spanning condition is that, for every \(N{\gt}L\), \(\ker H_L^{(N)}(B)=\operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \). This is the ground-space clause in [ 13 , Definition 3.9 ] . It is separate from the commutation equation \([\tau _j(P_L),P_L]=0\) and from the zero-energy equation for a single periodic MPS vector.
For a fixed chain length \(N\), this condition is the conjunction of the nearest-neighbor commutation equation, the zero-energy equation for the MPS vector, and the ground-space spanning equation:
The source statement in [ 13 , Theorem 3.10(iii) ] quantifies this condition over all \(N{\gt}2\).
This is the source quantification in [ 13 , Theorem 3.10(iii) ] . For every \(N{\gt}2\), the fixed-chain condition of Definition 12.1.16 holds:
This condition records the ground-space spanning equation, but does not prove it for a concrete canonical-form tensor.
The nearest-neighbor condition is the length-\(2\) specialization of the general commuting condition. The ground-vector condition combines this commuting equation with the frustration-free equation for \(V^{(N)}(A)\). For \(N\ge 2\), Lemma 11.4.47 with \(L=2\) gives \(h_iV^{(N)}(A)=0\) for every site \(i\). The fixed-chain ground-space condition also contains \(\ker H_2^{(N)}(B) = \operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \), and quantifying this condition over all \(N{\gt}2\) gives the nearest-neighbor case of the spanning condition above. Conversely, the all-chain condition is exactly the conjunction of the all-chain ground-vector equations and the nearest-neighbor instance of the spanning condition above. This is precisely the condition named in Definition 12.1.17. Since the displayed zero-energy equation follows from the parent-Hamiltonian construction for \(N{\gt}2\), independently of the commutation and spanning equations, this is equivalently
Assume \(N{\gt}0\) and \(L\le N\). Let \(\psi \) be a vector in the periodic chain ground space \(G_L^{(N)}(A)\). Then \(\psi \) satisfies all local zero-energy equations: for every site \(i\), one has \(h_i(A,L)\psi =0\).
Write \(R_{i,\eta }\psi \) for the cyclic length-\(L\) restriction of \(\psi \) at site \(i\), with the outside configuration fixed by \(\eta \). Membership in \(G_L^{(N)}(A)\) gives \(R_{i,\eta }\psi \in G_L(A)=\ker h_L(A)\) for every \(i\) and \(\eta \). Hence \(h_L(A)R_{i,\eta }\psi =0\). For a full configuration \(\sigma \), taking \(\eta =\sigma |_{\{ i,\ldots ,i+L-1\} ^c}\) gives
Thus \(h_i(A,L)\psi =0\) for every \(i\).
Let \(B\) be a canonical-form tensor with BNT components \(A_1,\ldots ,A_g\). If every component vector has zero energy for the parent Hamiltonian, \(H_L^{(N)}(B)V^{(N)}(A_j)=0\) for \(1\le j\le g\), then
It is enough to assume the local frustration-free equations \(h_iV^{(N)}(A_j)=0\) for every \(i\) and \(j\). For \(N{\gt}0\) and \(L\le N\), in the block-diagonal case \(B=\bigoplus _j\mu _jA_j\), with every \(\mu _j\ne 0\), the same inclusion follows because each local space \(G_L(A_j)\) is contained in \(G_L(B)\). This establishes the forward inclusion \(\operatorname{span}\{ V^{(N)}(A_j)\} _j \subseteq \ker H_L^{(N)}(B)\) of [ 13 , Definition 3.9 ] .
The kernel is a linear subspace. Hence it is enough to check the generators \(V^{(N)}(A_j)\). The first hypothesis gives this directly. Under the local frustration-free equations,
For a block-diagonal tensor, write \(R_{i,\tau }\) for the cyclic length-\(L\) restriction at site \(i\), with the outside configuration fixed by \(\tau \). Then \(R_{i,\tau }V^{(N)}(A_j) \in G_L(A_j)\subseteq G_L(B)\). The preceding theorem therefore gives \(h_i(B,L)V^{(N)}(A_j)=0\) for every \(i\).
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\). Suppose \(N{\gt}0\), \(L\le N\), and the periodic chain constraints for \(B\) split into the block constraints:
If every block chain space is contained in its periodic MPS line,
then the zero-energy space of the parent Hamiltonian is contained in the BNT vector span:
Theorem 12.3.14 gives \(\ker H_L^{(N)}(B)\subseteq \mathcal G_{N,L}(B)\). Together with the splitting hypothesis,
The blockwise containments give
and Lemma 13.1.2 gives
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\ne 0\). Suppose the physical alphabet and all block dimensions are nonempty. Assume that the block family is irreducible, left-canonical, has normalized self-overlap, and has no two blocks gauge-phase equivalent. Suppose also that each \(A_j\) is \(L_0\)-block-injective, that \(L_0{\gt}0\), and that \(\sum _a A^j_a(A^j_a)^\dagger =I \qquad (0\le j{\lt}r)\). Let \(N{\gt}0\), \(L{\gt}0\), \(L\le N\), \(L+L_0\le N\), and \((L_0+1) +(r-1)((L_0+1)+((L_0+1)+(L_0+1)))+1 \le L\). Assume the opened-boundary \(C^j,D^j\) comparison: whenever \(\psi \in \mathcal G_{N,L}(B)\) and \(\psi =\Gamma _N^B(\bigoplus _j X_j)\), there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every \(\rho \), and every \(\beta \), \(A^j_\beta C^j_{i,\rho } =((\mu _j^N X_j)A^j_\beta )A^j_\rho \). If every block chain space is contained in its periodic MPS line,
then
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\ne 0\). Suppose the physical alphabet and all block dimensions are nonempty. Assume that the block family is irreducible, left-canonical, has normalized self-overlap, and has no two blocks gauge-phase equivalent. Suppose also that each \(A_j\) is \(L_0\)-block-injective, that \(L_0{\gt}0\), and that \(\sum _a A^j_a(A^j_a)^\dagger =I \qquad (0\le j{\lt}r)\). Let \(N{\gt}0\), \(L{\gt}0\), \(L\le N\), \(L+L_0\le N\), and \((L_0+1) +(r-1)((L_0+1)+((L_0+1)+(L_0+1)))+1 \le L\). Assume that, for every boundary-crossing interval beginning at \(i\), the simultaneous products \(w\longmapsto (A^1_w,\ldots ,A^r_w), \qquad |w|=N-i\), span the product algebra \(\prod _jM_{D_j}(\mathbb {C})\). If every block chain space is contained in its periodic MPS line,
then
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\ne 0\). Then, for every \(N{\gt}L\), the source parent-Hamiltonian spanning condition \(\ker H_L^{(N)}(B) =\operatorname{span}\{ V^{(N)}(A_j):j=0,\ldots ,r-1\} \) is equivalent to the reverse containment, for the same \(N\):
In the nearest-neighbor case \(L=2\), the all-chain source condition of Definition 12.1.17 is equivalent to the conjunction
The forward direction of the displayed equality gives the containment by rewriting the kernel. Conversely, for \(N{\gt}L\), the forward containment gives
and the assumed reverse containment gives equality. For \(L=2\), the all-chain nearest-neighbor condition of Definition 12.1.17 decomposes as the conjunction
The second conjunct is the reverse containment just identified with the source spanning clause.
Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), where \(\mu _j\ne 0\). Suppose that, for every \(N{\gt}L\), \(G_L^{(N)}(B) =\bigvee _{j=0}^{r-1}\operatorname{span}\{ V^{(N)}(A_j)\} \). Then the source parent-Hamiltonian spanning equation holds: \(\ker H_N(B,L) =\operatorname{span}\{ V^{(N)}(A_j):j=0,\ldots ,r-1\} \). It is enough to prove the two equations
The finite-volume identity gives \(\ker H_N(B,L)=G_L^{(N)}(B)\). The assumed chain-space comparison therefore rewrites the kernel as \(\bigvee _{j=0}^{r-1}\operatorname{span}\{ V^{(N)}(A_j)\} \). The latter supremum is the BNT vector span. Conversely, Lemma 12.1.20 gives \(\operatorname{span}\{ V^{(N)}(A_j):j=0,\ldots ,r-1\} \subseteq \ker H_N(B,L)\), so the displayed equation is the source spanning clause. If the comparison is first stated as \(G_L^{(N)}(B)=\bigvee _jG_L^{(N)}(A_j)\), the single-block equations \(G_L^{(N)}(A_j)=\operatorname{span}\{ V^{(N)}(A_j)\} \) give the previous hypothesis.
A virtual similarity preserves the physical blocking relation and hence transfer idempotence. It also preserves every local MPS space, so the canonical parent interaction and all its translated local terms are unchanged.
Suppose \(B^i=XA^iX^{-1}\) for every physical letter. If the blocking relation for \(A\) is \(A^{i_1}A^{i_2} =\sum _j U^{j}_{i_1,i_2}A^j\), then conjugation by \(X\) gives the same relation for \(B\). Hence transfer idempotence is invariant under the similarity. At every length \(L\), the two tensors have the same local MPS space, so \(G_L(A)=G_L(B)\) and \(h_i^{(N)}(A)=h_i^{(N)}(B)\). The parent-interaction identities and the commutation equations follow.
If \(A\) has nonzero virtual dimension, is a normal tensor in the sense of [ 13 ] , and is a renormalization fixed point, then for every chain length \(N{\gt}2\) the two-site local terms commute. This theorem records the commutation part of the nearest-neighbor source condition, not the ground-space spanning part.
Put \(A\) into its trace-preserving gauge. This is a pure similarity, since the normal-tensor normalization fixes the transfer spectral radius at one. Similarity preserves the physical blocking relation and the local MPS spaces, hence also the canonical parent interactions. In the trace-preserving gauge, the Appendix B virtual-bond projectors give \(h_0^{(3)}h_1^{(3)} =h_1^{(3)}h_0^{(3)}\). Cyclic transport gives, for every \(N{\gt}2\) and every site \(i\), \(h_i^{(N)}h_{i+1}^{(N)} =h_{i+1}^{(N)}h_i^{(N)}\). For every pair \(i,j\) whose two-site supports are disjoint, locality gives \(h_i^{(N)}h_j^{(N)} =h_j^{(N)}h_i^{(N)}\). These cases exhaust all pairs of two-site terms on the periodic chain. Transporting the parent interactions back proves the claim.
If \(A\) has nonzero virtual dimension, is a normal tensor in the sense of [ 13 ] , and is a renormalization fixed point, then for every chain length \(N{\gt}2\) the MPS vector \(V^{(N)}(A)\) satisfies the commutation and zero-energy equations for nearest-neighbor local terms:
The zero-energy equation (6) is the parent-Hamiltonian frustration-freeness statement.
Let \(A\) have positive bond dimension and be a normal renormalization fixed-point tensor in the sense of [ 13 ] . If the canonical-form tensor consists of the single normal sector \(A\), then for every \(N{\gt}2\), \(\ker H_2^{(N)}(A) =\operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \). Its translated two-site interactions also commute and annihilate \(V^{(N)}(A)\). Hence the singleton family satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition.
Spectral normality gives eventual block injectivity. The fixed-point equation then implies that the one-site matrices already span the full matrix algebra. Thus \(A\) is injective. For \(N{\gt}2\), the finite-volume kernel identity and injective periodic-chain theorem give
The span of the matrix product vectors of the singleton family is the last space in this equality. Theorem 12.1.28 supplies commutation and zero energy on every chain of length greater than two.
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\) its translated two-site parent interactions commute pairwise. Thus \(B\) satisfies the nearest-neighbor commuting parent-Hamiltonian condition.
The fixed-point equation gives residual tensors \(U_j\) whose matrix-entry columns form one orthonormal family on the disjoint union of the sector matrix spaces. On the tensor square of this disjoint union, let \(Q\) be the sum of the sector bond projectors, extended by zero between unequal sectors. Write \(L_j\) and \(R_j\) for the left and right overlapping lifts of the sector-\(j\) bond projector. The single-sector relation and the mixed-sector orthogonality give
for \(j\ne k\), where (7) follows from the shared one-site factor \(E_j^*E_k=0\) for the canonical summand inclusions \(E_j\). Summing over the sectors shows that the two adjacent placements of \(Q\) commute.
If \(U\) is the common isometry, then transport gives the identity \((U\otimes U)Q(U^*\otimes U^*) =P_{G_2(B)}\), since the range on either side is \(\bigvee _j G_2(A_j)=G_2(B)\). Hence it is the complement of the canonical two-site parent interaction of \(B\). The two adjacent three-site parent terms therefore commute. Cyclic transport gives adjacent commutation on every chain with \(N{\gt}2\); disjoint local terms commute by locality, which proves the claim.
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\),
This establishes the multiplicity-one distinct-sector case of [ 13 , Theorem 3.10 ] .
The residual isometry equations imply that the simultaneous one-site matrices \((A_j^i)_j\) span the product of the block matrix algebras. Hence the same is true for every positive word length. The one-step restriction intersection therefore gives \(\mathcal G_{N,2}(B) \subseteq \bigvee _j G_N(A_j)\). Comparing this block decomposition before and after moving the periodic cut by one site shows that every block boundary matrix commutes with the corresponding one-site matrices. Thus each block component lies in \(\mathcal G_{N,2}(A_j)\), and \(\mathcal G_{N,2}(B) =\bigvee _j\mathcal G_{N,2}(A_j)\). Each block is a normal renormalization fixed point, so one-site injectivity and the injective periodic-chain theorem identify its chain space with the line spanned by \(V^{(N)}(A_j)\). The finite-volume kernel identity completes the proof.
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\) its translated two-site parent interactions commute and
Hence \(B\) satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition, establishing the multiplicity-one distinct-sector forward implication of [ 13 , Theorem 3.10 ] .
Theorem 12.1.30 gives pairwise commutation of the translated two-site interactions for every \(N{\gt}2\). The parent-Hamiltonian construction gives the zero-energy equation. Theorem 12.1.31 identifies its kernel with the span of the periodic matrix product vectors of the distinct sectors. These are the three clauses of the all-chain condition.
For a nonempty closed chain, the trace of the word \(A^{\sigma _0}\cdots A^{\sigma _{L-1}}\) is the sum over cyclic virtual-index configurations:
The trace is the diagonal sum:
For prescribed endpoints, the matrix product entry expands as
Substituting \(\alpha _L=\alpha _0\) in the trace gives the cyclic formula.
Suppose that \(A\) is equipped with Appendix B structural data \((X,\Lambda ,U)\): the matrix \(X\) is invertible, the diagonal entries of \(\Lambda \) are strictly positive, \(A^i=X\Lambda U^iX^{-1}\), and \(U\) satisfies the pair-index isometry equation below. Let \(\mathcal H_a\) and \(\mathcal H_b\) have bases indexed by \(\alpha ,\beta \in \{ 0,\ldots ,D-1\} \). The coefficients \(U^i_{\alpha ,\beta }\) define maps
and
Let \(A\) be equipped with Appendix B structural data \((X,\Lambda ,U)\) as in Definition 12.1.34. Its pair-index isometry equation
implies \(U^*U=1\). In particular, \(U\) is injective.
Expanding the coefficient of \(U^*Uv\) at \((\alpha ,\beta )\) and interchanging the two finite sums gives
A map with a left inverse is injective.
For every \(N\geq 0\), the coefficients of \(U\) define
with
The conjugate coefficient map is
This is the tensor power occurring in the basic-vector expression [ 13 , Equation (3.17) ] .
For every \(N\geq 0\), one has \(U^{*\otimes N}U^{\otimes N}=1\). Hence \(U^{\otimes N}\) is injective.
For virtual-pair configurations \(p\) and \(q\), interchange the two finite sums and factor the sum over physical configurations site by site:
The pair-index isometry equation turns the last product into \(\prod _t\delta _{p_t,q_t}=\delta _{p,q}\). Therefore, the coefficient at \(p\) is \(v_p\). A map with a left inverse is injective.
Let \(A^i=X\Lambda U^iX^{-1}\) be the Appendix B structural form, and write \(A_{\mathrm{core}}^i=\Lambda U^i\). The canonical local parent interaction is unchanged by this basis change: \(q_L(A)=q_L(A_{\mathrm{core}})\).
The basis change gives a gauge equivalence between \(A_{\mathrm{core}}\) and \(A\), hence equality of their length-\(L\) local ground spaces. The parent interaction is the orthogonal complement projector of this local ground space.
For Appendix B structural data, the two-site basic support is \(G_2(A_{\mathrm{core}})=G_2(\Lambda U)\).
Put \(|\varphi \rangle =\sum _b\lambda _b|b,b\rangle \). For outer-boundary coefficients \(v_{a,c}\), insertion of this vector on the middle virtual bond is the map \(I_\varphi \) defined by
The two-site physical map is \(U^{\otimes 2}I_\varphi \). A boundary matrix \(Y\) determines outer-boundary coefficients by \((v_Y)_{a,c}=\lambda _aY_{c,a}\). Opening the outer virtual bond in the two-site contraction of [ 13 , Equations (3.17)–(3.18) ] gives these coefficients.
Write \(s=\langle \varphi ,\varphi \rangle =\sum _b\lambda _b^2\). The orthogonal projector onto \(\mathbb {C}\varphi \) has pair-index coefficients
On three virtual site pairs, let \(P_{01}\) place this projector on the bond joining the right index of site zero to the left index of site one, and let \(P_{12}\) place it on the next bond.
The two adjacent placements commute: \(P_{01}P_{12}=P_{12}P_{01}\). The following support-location schematic marks the two bonds on which the projectors act; it does not depict the doubled operator network for the equality above.
![\begin{tenkz}[rows={wire}, cols=5, west=open, east=open, physical=up]
\tn[skin=box]{U_0} &
\tn[physical=none, label pos=s]{\varphi_{01}} &
\tn[skin=box]{U_1} &
\tn[physical=none, label pos=s]{\varphi_{12}} &
\tn[skin=box]{U_2}
\tndeclare{species}{support}{hue=source:red}
\tnmark[form=enclosure, species=support, tint, label pos=s]
{(1,2)}{$\supp(P_{01})$}
\tnmark[form=enclosure, species=support, tint, label pos=s]
{(1,4)}{$\supp(P_{12})$}
\end{tenkz}](tenkz_svg/tenkz-783c6927ca8e52d6.svg)
The contracted bond coordinates are disjoint. Thus replacing the first bond by the diagonal basis vector \(|k,k\rangle \) and the second by \(|\ell ,\ell \rangle \) is independent of the order. Expanding both rank-one projectors gives the same double sum over \((k,\ell )\).
Strict positivity of the \(\lambda _a\) makes \(Y\mapsto v_Y\) surjective.
For prescribed coefficients \(v_{a,c}\), take \(Y_{c,a}=\lambda _a^{-1}v_{a,c}\).
Opening the outer virtual bond in the two-site contraction of [ 13 , Equations (3.17)–(3.18) ] gives the following local-support family. For every boundary matrix \(Y\), \(U^{\otimes 2}I_\varphi (v_Y) = \Gamma _2(\Lambda U)(Y)\). Consequently, \(\operatorname{ran}(U^{\otimes 2}I_\varphi )=G_2(\Lambda U)\).
At a two-site physical configuration \((i,j)\), the bond insertion gives \(\sum _{a,b,c}U^i_{a,b}U^j_{b,c}\lambda _b\lambda _aY_{c,a}\). Expanding \(\operatorname{tr}((\Lambda U^i)(\Lambda U^j)Y)\) gives the same sum, so the two maps agree after the weighted boundary parametrization. Since that parametrization is surjective, their ranges are equal.
Let \(P_2\) be the orthogonal projector onto \(G_2(\Lambda U)\), as in the local parent-space construction of [ 13 , Definition 3.9 ] .
The projector \(P_2\) satisfies
In particular, \(q_2(\Lambda U)\) annihilates the image of \(U^{\otimes 2}I_\varphi \), while \(P_2\) fixes that image pointwise.
Orthogonal projection onto \(G_2(\Lambda U)\) has that subspace as its range. The preceding theorem identifies this range with \(\operatorname{ran}(U^{\otimes 2}I_\varphi )\). Orthogonal projection onto the complementary subspace is \(q_2(\Lambda U)\), hence \(P_2=1-q_2(\Lambda U)\). Since \(q_2(\Lambda U)^2=q_2(\Lambda U)\),
If \(\psi \) lies in \(\operatorname{ran}(U^{\otimes 2}I_\varphi )\), then \(\psi \in G_2(\Lambda U)=\ker q_2(\Lambda U)\), and therefore
Let \(\mathcal V=\bigsqcup _j\{ j\} \times [D_j]\times [D_j]\) be the disjoint union of the within-sector virtual pairs, and let \(V\colon \mathcal V\to \mathbb {C}^d\) be the common physical map whose matrix elements are \(V_{i,(j,\alpha ,\beta )}=(U_j^i)_{\alpha ,\beta }\). For each sector insert the vector \(|\varphi _j\rangle =\sum _b\lambda _{j,b}|b,b\rangle \) on the middle virtual bond, apply \(V\) on both sites, and sum the resulting physical vectors. This defines the equal-sector two-site embedding
Unequal-sector virtual bonds do not occur. The corresponding physical support operator \(P_{2,\mathrm{res}}\) is the orthogonal projector onto \(G_2(\bigoplus _j B_j)\).
The common physical map is an isometry, \(V^*V=1\). Moreover, the equal-sector bond embedding and the physical support projector satisfy
and \(P_{2,\mathrm{res}} = 1-q_2\! \left(\bigoplus _j B_j\right)\). If the \(B_j\) form a basis of normal tensors and their direct sum is a renormalization fixed point, the sectorwise Appendix B decompositions and the common residual isometry required above exist.
The isometry relation \(V^*V=1\) is the one-letter Gram identity on the disjoint union of the within-sector virtual pairs. In each sector, the bond insertion has range \(G_2(B_j)\); allowing independent outer-boundary coefficients and summing over sectors therefore gives \(\sum _jG_2(B_j)\). The local ground space of a direct sum is this sum, which proves (8) and (9). Orthogonal projection onto that space is complementary to its parent interaction. Finally, the residual-isometry form of the cross-block fixed-point structure supplies the simultaneous sector decompositions.
Let \((P_2)_{01}\) and \((P_2)_{12}\) be the two adjacent placements of the two-site basic support projector on three physical sites. Then \((P_2)_{01}(P_2)_{12} =(P_2)_{12}(P_2)_{01}\). This is the local three-site consequence of the parent-space construction and the basic-vector form in [ 13 , Definition 3.9, Equations (3.16)–(3.18), and Appendix B ] .
The one-site map \(U\) is an isometry. The preceding range identity therefore identifies the physical projector with the isometric transport of the virtual bond projector: \(P_2 =U^{\otimes 2}P_\varphi (U^*)^{\otimes 2}\). On three sites, multiplication of the two adjacent transports cancels the middle factors by \(U^*U=1\). Consequently their products are
respectively. They agree by Theorem 12.1.42. The following support-location schematic records the one-site embedding and the two overlapping physical supports. It does not depict the doubled \(U\cdots U^*\) operator networks in the identities above.
![\begin{tenkz}[rows={wire}, cols=5, west=open, east=open, physical=up]
\tn[skin=box]{U_0} &
\tn[physical=none, label pos=s]{\varphi_{01}} &
\tn[skin=box]{U_1} &
\tn[physical=none, label pos=s]{\varphi_{12}} &
\tn[skin=box]{U_2}
\tndeclare{species}{support}{hue=source:red}
\tndeclare{species}{transported}{hue=source:blue}
\tnmark[form=enclosure, species=support, tint, label pos=s]
{(1,2)}{$\supp(P_{01})$}
\tnmark[form=enclosure, species=support, tint, label pos=s]
{(1,4)}{$\supp(P_{12})$}
\tnmark[form=enclosure, species=transported, label pos=nw]
{(1,1) .. (1,3)}{$\supp((P_2)_{01})$}
\tnmark[form=enclosure, species=transported, label pos=ne]
{(1,3) .. (1,5)}{$\supp((P_2)_{12})$}
\end{tenkz}](tenkz_svg/tenkz-f134022167074485.svg)
The \(AX\) coefficient-space representative is, before the \(AX\)-lift, the canonical two-site parent interaction of the core tensor, \(\widehat Q_{AX}=q_2(A_{\mathrm{core}})\). The source projector \(Q_{AX}\) of Definition D.2 acts on \(\mathcal H_A\otimes \mathcal H_X\). The displayed operator is the two-site coefficient representative that will be placed on the \(AX\) face; it is not by itself the construction of that source projector. The \(XB\) coefficient-space representative is the same operator, \(\widehat Q_{XB}=\widehat Q_{AX}\), later placed on the \(XB\) face; the source projector \(Q_{XB}\) acts on \(\mathcal H_X\otimes \mathcal H_B\).
On the homogeneous three-single-site coefficient space \((\mathbb {C}^d)^{\otimes 3}\), the hatted Appendix B (AX) representative acts as \([\widehat Q_{AX}]_{AX} =[\widehat Q_{AX}]\otimes I_d\). This equation gives the coefficient-factor specialization of the left local action in [ 13 , Definition D.2 ] . It does not construct a projector for an arbitrary factorization \(\mathcal H_A\otimes \mathcal H_X\otimes \mathcal H_B\).
Apply the left identity in Theorem 12.1.3 to \(\widehat Q_{AX}\).
On the homogeneous three-single-site coefficient space \((\mathbb {C}^d)^{\otimes 3}\), the hatted Appendix B (XB) representative acts as \([\widehat Q_{XB}]_{XB} =I_d\otimes [\widehat Q_{XB}]\). This equation gives the coefficient-factor specialization of the right local action in [ 13 , Definition D.2 ] . It does not construct a projector for an arbitrary factorization \(\mathcal H_A\otimes \mathcal H_X\otimes \mathcal H_B\).
Apply the right identity in Theorem 12.1.3 to \(\widehat Q_{XB}\).
As coefficient-space representatives, \(\widehat Q_{AX}=q_2(A)\) and \(\widehat Q_{XB}=q_2(A)\).
Both operators are \(q_2(A_{\mathrm{core}})\), and the Appendix B basis change leaves \(q_2\) unchanged.
The two Appendix B two-site coefficient representatives are idempotent:
By the preceding identification, both representatives are the canonical two-site parent interaction \(q_2(A)\). The latter is an orthogonal projector, hence is idempotent.
The two Appendix B coefficient representatives satisfy
and their adjacent actions on the common three-site coefficient space commute:
Thus they satisfy the algebraic overlapping two-site condition of Definition 12.1.2, providing the coefficient-space analog of the commutation condition in [ 13 , Definition D.2 ] .
Both coefficient representatives are the complement \(1-P_2\) of the two-site basic support projector. Their idempotency was established in Lemma 12.1.54. Since the two adjacent lifts of \(P_2\) commute, their complementary lifts commute as well.
If \(A\) is injective, then \(G_3(A) = \ker (q_2(A)_{AX}) \cap \ker (q_2(A)_{XB})\). More generally, on \(L+1\) sites, simultaneous annihilation by the two adjacent length-\(L\) parent interactions is equivalent to membership in \(G_{L+1}(A)\) whenever \(A\) is injective and \(L{\gt}1\).
Let \(A\) be injective and let an Appendix B structural datum be fixed. Put
with the two copies of \(q_2(A)\) lifted to the indicated faces of the three-site coefficient space. Then
These are the coefficient-space analogs of the idempotence, commutation, and kernel-intersection equations in [ 13 , Definition D.2 ] . The hatted operators are not identified here with the source projectors on \(\mathcal H_A\otimes \mathcal H_X\) and \(\mathcal H_X\otimes \mathcal H_B\). For a normal left-canonical RFP tensor, injectivity follows from the RFP structural theorem and is not an additional hypothesis.
Suppose the first two length-two parent interactions commute on the three-site chain: \(h_0^{(3)}h_1^{(3)} =h_1^{(3)}h_0^{(3)}\). Then, for every \(N{\gt}2\) and every \(i\in \mathbb Z/N\mathbb Z\), the two adjacent translates commute: \(h_i^{(N)}h_{i+1}^{(N)} =h_{i+1}^{(N)}h_i^{(N)}\). For each exterior configuration \(\tau \), let \(R_{i,\tau }^{(3)}\) restrict to the cyclic three-site window beginning at \(i\). It intertwines the two \(N\)-site terms with the two three-site terms:
Thus the same local \(A\! X\! B\) calculation is placed at every cyclic translate through the two restriction identities above. The hypothesis \(N{\gt}2\) is essential to this transport: at \(N=2\) the two cyclic windows contain the same sites in opposite orders. This case is not part of [ 13 , Theorem 3.10 ] .
Restrict an \(N\)-site vector to the cyclic window \((i,i+1,i+2)\) while holding all other physical indices fixed. Window extraction and replacement identify the two restricted local terms with \(h_0^{(3)}\) and \(h_1^{(3)}\). Apply the three-site commutator to the restricted vector and then evaluate at the original three-site configuration. Replacing a window by the values extracted from that configuration recovers the original \(N\)-site configuration.
For every Appendix B structural datum and every periodic chain of length \(N{\gt}2\), the adjacent translated length-two parent interactions commute.
A normal left-canonical renormalization fixed-point tensor of nonzero virtual dimension has pairwise commuting length-two parent interactions on every periodic chain of length \(N{\gt}2\), and its periodic matrix product vector is annihilated by every such interaction. The removal of the left-canonical hypothesis and the ground-space spanning equation in [ 13 , Definition 3.9 ] remain separate.
Suppose the lifted \(AX\) and \(XB\) representatives commute:
Then \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) satisfy the algebraic overlapping two-site condition of Definition 12.1.2.
Let \(Q_{AX}\) and \(Q_{XB}\) be the two-site coefficient-space transports of source projectors satisfying the Definition D.2 local condition on \(A X B\). If these transported projectors agree with the Appendix B coefficient representatives,
then \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) satisfy the algebraic overlapping two-site condition of Definition 12.1.2. Consequently, \(h_0^{(3)}(A,2)h_1^{(3)}(A,2) = h_1^{(3)}(A,2)h_0^{(3)}(A,2)\).
Rewriting the two projectors in the Definition D.2 condition as \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) gives the algebraic overlapping condition. The three-site commutation is then Lemma 12.1.64.
On the three-site \(A,X,B\) window, the two adjacent length-two local terms are the two local lifts
Apply the three-site local-term identities to \(q_2(A)\), and replace \(q_2(A)\) by \(\widehat Q_{AX}\) or \(\widehat Q_{XB}\) using the preceding two-site identifications.
If the Appendix B \(AX\) and \(XB\) coefficient representatives satisfy the overlapping two-site commutation datum
then the corresponding first two translated length-two local terms on the three-site window commute: \(h_0^{(3)}(A,2)h_1^{(3)}(A,2) = h_1^{(3)}(A,2)h_0^{(3)}(A,2)\).
By Lemma 12.1.63, the two translated local terms are exactly the \(AX\) and \(XB\) lifts of \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\). The displayed equality is therefore the commutation clause of the overlapping two-site datum.
If the Appendix B \(AX\) and \(XB\) coefficient representatives satisfy the lifted Definition D.2 commutator
then the corresponding first two translated length-two local terms on the three-site window commute: \(h_0^{(3)}(A,2)h_1^{(3)}(A,2) = h_1^{(3)}(A,2)h_0^{(3)}(A,2)\). The same conclusion holds if one supplies a Definition D.2-shaped local condition whose two operators are these coefficient-space representatives, rather than the source projectors themselves.
The lifted commutator, together with the Appendix B idempotency already recorded above, gives the overlapping two-site commutation datum. A supplied Definition D.2-shaped local condition on the coefficient representatives implies the same conclusion by retaining its commutation clause and not using the kernel-intersection condition
The Appendix B structural form supplies \(A^i=X\Lambda U^iX^{-1}\). In the source unit pair-index convention, the residual tensor satisfies
Therefore the residual core tensor \(\Lambda U^i\) satisfies the weighted pair-index relation
This is an orthogonality relation after summing over the physical letter \(i\). It is not a coefficient identity for a fixed physical word \(\sigma \). The coefficient associated to the source basic-vector formula is
This is the coefficient expression for the source vector
where \(\varphi _j\) is shared by \(b_t\) and \(a_{t+1}\) and \(U\) acts on \((a_t,b_t)\). At length two, this cyclic coefficient is the structural two-site amplitude: \(C_2(\sigma )=\phi _2(\sigma )\). The change of basis does not alter either the local spaces \(G_L\) or the periodic MPS coefficients: \(G_L(A)=G_L(\Lambda U)\). It therefore also leaves the canonical parent interaction unchanged: \(q_L(A)=q_L(\Lambda U)\). At length two, the Appendix B two-site basic support is \(G_2(\Lambda U)\), and the coefficient-space representatives \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) are the two copies of \(q_2(\Lambda U)\) before they are lifted to the \(AX\) and \(XB\) faces. Hence
as coefficient-space representatives. On the three-site window their two lifts give the local identifications
Consequently, for every chain length \(L\geq 1\), the original tensor has the same cyclic coefficient formula: \(V^{(L)}(A)_\sigma =C_L(\sigma )\). In particular, the corresponding two-site amplitude is
The source places each virtual pair across neighboring sites and applies an on-site physical isometry, as in [ 13 , equations (3.17)–(3.18) and Appendix B ] . On a three-site chain, the two adjacent length-two local terms are the \(AX\) and \(XB\) actions of the canonical two-site parent interaction. Theorem 12.1.55 constructs the adjacent physical support projectors from the basic-vector expression and proves the overlapping relation
Theorem 12.1.58 transports this identity around a periodic chain. Consequently, for every \(N{\gt}2\) and every pair of sites \(i,j\in \mathbb {Z}/N\mathbb {Z}\), \(h_i(A,2)h_j(A,2) = h_j(A,2)h_i(A,2)\). Lemma 12.1.13 gives this pairwise commutation directly from adjacent commutation. Together with Lemma 11.4.47, this gives the zero-energy equation for \(V^{(N)}(A)\). Neither conclusion requires a disjoint physical-pair factorization. The full source implication additionally includes the ground-space spanning equation. The coefficient-space analog of the local kernel-intersection equation is Theorem 12.1.57; identifying its hatted maps with the source projectors remains separate and is not needed for the chain commutator.
The unit pair-index isometry and cyclic coefficient formula in Remark 12.1.66 do not imply factorization of a fixed physical-word coefficient over disjoint physical pairs. For example, take virtual indices \(0,1\), physical labels \((a,b)\), and \(U^{(a,b)}_{x,y}=\delta _{x,a}\delta _{y,b}\), with \(A^{(a,b)}=\lambda _a E_{a,b}\) and \(\lambda _0,\lambda _1{\gt}0\). The pair-index isometry holds. For the word \(\sigma =((0,1),(1,0),(1,0),(0,1))\), the cyclic coefficient is
In contrast, the product over disjoint physical pairs is
Let \(P\) be a sector decomposition with representatives \(A_1,\ldots ,A_g\) and total tensor \(B\). Suppose that the vectors \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\) are linearly independent for all sufficiently large \(N\), and that \(B\) satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition with these representatives. Then there is \(N_0\) such that, for every \(N{\gt}N_0\), \(\dim _{\mathbb {C}}\ker H_2^{(N)}(B) = g\). Thus the ground-state degeneracy is eventually constant, as required in [ 5 , Section IV ] .
Choose \(N_0\geq 2\) beyond the linear-independence threshold. For \(N{\gt}N_0\), the ground-space spanning equation gives \(\ker H_2^{(N)}(B) = \operatorname{span}\{ |V^{(N)}(A_j)\rangle :j=1,\ldots ,g\} \). Identifying these vectors with their coefficient functions preserves linear independence. The dimension of their span is therefore \(g\).
Let \(A\) be a matrix product tensor. Write its canonical two-site parent interaction in the ordered-pair basis of the two-site physical Hilbert space, and denote the resulting matrix by \(Q_A\).
Suppose that the canonical length-two parent terms of \(A\) commute on a three-site chain. Then \(Q_A\) is Hermitian, and its natural placements on the first and last pairs of a three-fold tensor product commute:
The parent interaction is the orthogonal projector onto the orthogonal complement of the local matrix product space, hence is Hermitian. Transporting the adjacent three-site parent-term commutator to the common left-associated tensor-product basis gives the displayed identity.
Under the preceding hypothesis, there are positive integers \(d_q,m_q\), a unitary identification
and Hermitian matrices \(R_q\) on \(\mathcal H\otimes \mathcal H_{q,l}\) and \(S_q\) on \(\mathcal H_{q,r}\otimes \mathcal H\) such that, after conjugating the middle site by this unitary, the interaction decomposes as
on the first pair and
on the last pair. This is the spatial decomposition of [ 5 , Lemma 2.1 ] for the canonical parent interaction.
Apply the finite-dimensional spatial decomposition for two Hermitian overlapping operators to \(Q_A\) in both positions, using Theorem 12.1.70.
Suppose that a one-site unitary identifies the physical Hilbert space with a finite direct sum
and that in these coordinates the complementary two-site interaction is \(\bigoplus _{a,b}\mathbb {1}_{a,l}\otimes P_{a,b}\otimes \mathbb {1}_{b,r}\), where each \(P_{a,b}\) is an orthogonal projector. The directed edge \(a\to b\) is present precisely when \(\operatorname{ran}P_{a,b}\) is nonzero. An ordered cycle of length \(N\) is a sequence \((a_0,\ldots ,a_{N-1})\) for which every cyclically adjacent edge \(a_n\to a_{n+1}\) is present. Cyclic rotations are not identified. This is the graph following equation (2) of [ 5 , Section III ] .
Let \(B\geq 0\) be a two-site operator whose overlapping placements commute: \((B\otimes \mathbb {1})(\mathbb {1}\otimes B) = (\mathbb {1}\otimes B)(B\otimes \mathbb {1})\). There are positive integers \(d_{q,l},d_{q,r}\), a one-site unitary \(U\), and positive semidefinite operators \(\eta _{q,h}\) on \(\mathcal H_{q,r}\otimes \mathcal H_{h,l}\) such that
This is the symmetric two-sided form obtained from [ 5 , Lemma 2.1 and Section III, equation (2) ] .
Apply the spatial decomposition to the two overlapping placements of \(B\). In the resulting coordinates, matrix entries vanish unless both the left and right sector labels agree. For fixed sectors \(q,h\), choose unit vectors \(l_0\in \mathcal H_{q,l}\) and \(r_0\in \mathcal H_{h,r}\) and define
The left block action makes the right outer factor an identity, while the right block action makes the left outer factor an identity. Hence the displayed matrix element is independent of the chosen outer coordinates and gives the stated direct sum. Each \(\eta _{q,h}\) is a compression of the positive semidefinite matrix \(\widetilde B\), so it is positive semidefinite.
Suppose that the canonical length-two parent terms of \(A\) commute on a three-site chain. Then the complementary two-site interaction admits a finite sector graph as in Definition 12.1.72.
The canonical two-site parent interaction \(Q_A\) is an orthogonal projector. Thus \(\mathbb {1}-Q_A\) is a positive semidefinite orthogonal projector. The overlapping placements of \(\mathbb {1}-Q_A\) commute because those of \(Q_A\) commute. Apply Theorem 12.1.73 to obtain
The left-hand side is idempotent, so every block on the right is idempotent. Evaluating the two outer identity factors at fixed unit vectors gives \(\eta _{q,h}^2=\eta _{q,h}\). Since \(\eta _{q,h}\geq 0\), each \(\eta _{q,h}\) is an orthogonal projector and therefore defines the required edge projection.
For every \(N\geq 2\), the periodic ground-space dimension is
where the sum is over ordered cycles and the indices are read modulo \(N\). This is equations (3)–(4) of [ 5 , Section III ] for chains of length at least two. The one-site periodic interaction requires a separate convention.
The translated parent interactions commute. Hence the ordered product of their complementary orthogonal projectors is the projector onto \(\ker H_N\). In the cyclic sector coordinates this product is block diagonal, with the block indexed by \((a_0,\ldots ,a_{N-1})\) equal to the tensor product of the adjacent projectors \(P_{a_n,a_{n+1}}\). Taking the trace gives the product of their ranks. A sequence containing a missing edge contributes zero, leaving exactly the displayed ordered-cycle sum.
At period two, the ground-space dimension is
The sum is over ordered pairs: if \(a\ne b\), then \((a,b)\) and \((b,a)\) are distinct summands. This is the length-two convention stated after equation (4) of [ 5 , Section III ] .
Specialize the ordered-cycle formula to \(N=2\). Its two cyclic factors are \(P_{a,b}\) and \(P_{b,a}\), which gives the displayed product of ranks.
Under the convention that an \(L\)-site interaction vanishes for \(L{\gt}N\), the two-site parent Hamiltonian on a one-site periodic chain satisfies
Since \(2{\gt}1\), the interaction vanishes by convention, and the kernel of the zero operator is the full one-site Hilbert space.
Let \(G\) be a finite directed graph whose edge weights are non-negative integers, and let
be the weighted sum over ordered cyclic vertex sequences. Suppose that \(s_N\) is constant for all sufficiently large \(N\). Then every edge that belongs to a directed cycle has weight one, every directed cycle is a repetition of a single loop, and, for every \(N{\gt}0\), \(s_N=\# \{ a:w(a,a){\gt}0\} \). This is the comparison in [ 5 , Section IV ] .
Choose a sufficiently large \(N\) such that the sums at lengths \(N\), \(N+1\), and \(N(N+1)\) have their common eventual value. Repeating each ordered \(N\)-cycle \(N+1\) times gives an injection into the cycles of length \(N(N+1)\), while repeating each ordered \((N+1)\)-cycle \(N\) times gives a second injection. Comparison of the three weighted sums shows first that every positive cyclic weight is one. Both injections must then be exhaustive. Hence a cycle at the common length has periods \(N\) and \(N+1\); comparison of adjacent entries gives period one. Repeating an arbitrary shorter cycle to a sufficiently large length proves that it is constant. The only surviving ordered cycles at any positive length are therefore the constant words supported on positive loops, each of weight one.
If the parent-ground-space dimension is eventually constant, then for every \(N{\gt}2\) it equals the number of positive loops in the sector graph. In particular, this conclusion holds under the BNT and all-chain nearest-neighbor parent-ground-space hypotheses of Theorem 12.1.68.
The ordered-cycle dimension formula identifies the eventual parent-ground-space dimension with the eventual weighted-cycle sum. The preceding theorem identifies that sum, at every positive length, with the loop number. Transport between coefficient and Euclidean coordinates preserves the dimension of the ground space.
Let \(a\to a\) be a positive loop and choose a nonzero vector \(\varphi _a\in \operatorname{ran}P_{a,a}\). In sector coordinates define the matrix product tensor \((C_a^{(r,l)})_{x,y}=\delta _{x,l}\varphi _a(r,y)\) and set it to zero outside sector \(a\). The physical loop tensor is the image of \(C_a\) under the one-site unitary of the spatial decomposition. Its associated cyclic product pairs the right factor at site \(n\) with the left factor at site \(n+1\). The corresponding \(N\)-site product state is denoted by \(\Psi _a^{(N)}\in (\mathbb {C}^d)^{\otimes N}\).
Let \(\Phi _a=(\varphi _a(r,l))_{r,l}\) and write \(r_a=\operatorname{rank}\Phi _a\). Choose a rank factorization
The tensor on the minimal Schmidt support is
again extended by zero outside sector \(a\). Its physical tensor is its image under the same one-site unitary.
Let \(\nu _a=\lVert \varphi _a\rVert _2{\gt}0\), where the norm is the Hilbert-space norm of the bond vector, or equivalently the Frobenius norm of \(\Phi _a\). In either coordinate system, the normalized representative is \(\widehat C_a=\nu _a^{-1}\widetilde C_a\). The raw coordinate and physical tensors are retained separately.
The Schmidt rank \(r_a\) of the nonzero loop bond vector is positive.
If \(r_a=0\), then the intermediate index set in the factorization \(\Phi _a=X_aY_a\) is empty. Every entry of \(X_aY_a\) is therefore zero, contrary to \(\varphi _a\ne 0\).
The tensor \(\widetilde C_a\) on the Schmidt support is injective, as is its image under the one-site unitary. In particular, the physical tensor is normal.
In a rank factorization \(\Phi _a=X_aY_a\) through \(r_a=\operatorname{rank}\Phi _a\), the columns of \(Y_a\) and the rows of \(X_a\) each span \(\mathbb {C}^{r_a}\). Their pairwise outer products therefore span \(M_{r_a}(\mathbb {C})\), and these outer products are precisely the loop-sector letters of \(\widetilde C_a\). A unitary change of physical coordinates preserves this span. One-site injectivity implies normality.
The transfer map of the unnormalized coordinate tensor is
Hence the transfer map of \(\widehat C_a\) is idempotent. This remains true after the physical unitary rotation. The normalized physical tensor is injective and therefore normal.
Summing the Kraus terms over the loop-sector coordinates gives the stated rank-one formula. Cyclicity of the trace and \(\Phi _a=X_aY_a\) give \(\operatorname{tr}(X_a^*X_aY_aY_a^*)=\operatorname{tr}(\Phi _a^*\Phi _a)=\nu _a^2\). Scaling every letter by \(\nu _a^{-1}\) scales the transfer map by \(\nu _a^{-2}\), which makes it idempotent. A unitary change of physical coordinates leaves the transfer map invariant, and the nonzero scalar preserves injectivity.
For every \(N{\gt}0\), the periodic matrix product vector of \(C_a\) has coefficient \(\prod _{n=0}^{N-1}\varphi _a(r_n,l_{n+1})\) on a cyclic configuration in sector \(a\), and coefficient zero on every other sector configuration. Consequently, the periodic vector of the physical loop tensor is exactly the sitewise-unitary image of this cyclic product-of-pairs state. The same identities hold for the tensor on the minimal Schmidt support. For the normalized minimal tensor it is \(\nu _a^{-N}\Psi _a^{(N)}\).
Write \(D_a^{\mathrm L}\) for the left-factor dimension of sector \(a\). Expanding the closed matrix product on a sector configuration \(((r_n,l_n))_{n=0}^{N-1}\) gives
where all site labels are read cyclically. Indeed, the Kronecker deltas leave only the assignment \(g_n=l_n\).
For the minimal tensor, let \(S^{(r,l)}\) be the rectangular matrix whose only nonzero row is row \(l\), equal there to row \(r\) of \(X_a\). Then \(C_a^{(r,l)}=S^{(r,l)}Y_a\) and \(\widetilde C_a^{(r,l)}=Y_aS^{(r,l)}\). For every nonempty word there is therefore a rectangular matrix \(R\) such that its two products are \(RY_a\) and \(Y_aR\), respectively. Cyclicity of the trace gives \(\operatorname{tr}(RY_a)=\operatorname{tr}(Y_aR)\), proving equality of all positive-length periodic vectors.
If \(A_a\) denotes the physical loop tensor and \(U\) the one-site unitary, then for a physical configuration \(s=(s_0,\ldots ,s_{N-1})\) one has
Substitution of the preceding cyclic coefficient formula gives the stated sitewise-unitary image.
Suppose that \(A\) is equipped with Beigi sector data, with one-site spatial unitary \(U\). For \(N\geq 2\), let \(Q_N\) be the product of the translated complementary parent interactions and let \(\widetilde Q_N\) be the corresponding product in sector coordinates. Then \((U^*)^{\otimes N}Q_NU^{\otimes N} = \widetilde Q_N\).
The two-site coordinate identity gives \((U^*\otimes U^*)(\mathbb {1}-h)(U\otimes U)=\widetilde Q\) for one bond. Conjugation by \((U^*)^{\otimes N}\) carries every translated copy of this bond identity to the corresponding translated sector interaction. Since unitary conjugation distributes over an ordered product,
where \(Q_n\) and \(\widetilde Q_n\) are the translated physical and sector complementary interactions, respectively.
Suppose that \(A\) is equipped with Beigi sector data. For \(N\geq 2\), let \(h_0,\ldots ,h_{N-1}\) be its translated length-two parent interactions and set \(Q_N=\prod _{n=0}^{N-1}(\mathbb {1}-h_n)\). Every vector fixed by \(Q_N\) belongs to the kernel of the periodic parent Hamiltonian \(H_N=\sum _n h_n\).
Under the Euclidean identification \(E\), the matrix product of the complementary interactions is carried to
The operators \(h_n^{\mathrm{E}}\) are orthogonal projections. Beigi sector data supply pairwise commutativity, which Euclidean transport preserves. For all \(0\leq m,n{\lt}N\), one has \(h_m^{\mathrm{E}}h_n^{\mathrm{E}} =h_n^{\mathrm{E}}h_m^{\mathrm{E}}\). For any commuting orthogonal projections \(P_0,\ldots ,P_{N-1}\),
Applying this identity to the \(h_n^{\mathrm{E}}\) gives
Thus \(Q_Nv=v\) implies \(E^{-1}v\in \ker H_N^{\mathrm{E}}\), and conjugating back by \(E\) gives \(v\in \ker H_N\).
Let \(a\to a\) be a positive loop. For every \(N\geq 2\), its physical cyclic product-of-pairs state belongs to the kernel of the length-two periodic parent Hamiltonian. Equivalently, its Euclidean representative belongs to the parent ground space.
In sector coordinates, the complete product \(\widetilde Q_N\) of the complementary two-site interactions is block diagonal over cyclic sector sequences. Write \(\Phi _a(k;x)\) for the coefficient of the loop cyclic product in the block indexed by \(k\). Its support lies in sector \(a\), so for every \(k\ne (a,\ldots ,a)\) and every edge-index tuple \(x\), one has \(\Phi _a(k;x)=0\). Thus only the constant cyclic sector sequence contributes. On this block, \(\widetilde Q_N\) is the tensor product of the edge projectors. Since \(\varphi _a\in \operatorname{ran}P_{a,a}\),
If \(U\) is the one-site spatial unitary and \(\Psi _a=U^{\otimes N}\Phi _a\) is the physical loop state, unitary conjugation gives
Theorem 12.1.88 now yields \(\Psi _a\in \ker H_N\).
Suppose that the parent-ground-space dimension is eventually constant. For every \(N{\gt}2\), the positive-loop product states form a nonzero pairwise orthogonal family and
In particular, the same conclusion holds under the BNT and all-chain nearest-neighbor parent-ground-space hypotheses.
Write \(\Phi _a^{(N)}\) for the cyclic product in sector coordinates and \(\Psi _a^{(N)}=U^{\otimes N}\Phi _a^{(N)}\) for the physical state, where \(U\) is the one-site spatial unitary. Choose \(q\) with \(\varphi _a(q)\ne 0\). On the constant edge-index configuration,
Thus \(\Phi _a^{(N)}\ne 0\), and unitarity gives \(\Psi _a^{(N)}=U^{\otimes N}\Phi _a^{(N)}\ne 0\). If \(a\ne b\), then every sector configuration \(s\) satisfies \(\Phi _a^{(N)}(s)=0\) or \(\Phi _b^{(N)}(s)=0\), and hence
Moreover,
Thus the loop states are linearly independent. Their span is contained in \(\ker H_N\) by Theorem 12.1.89. Its dimension is the number of positive loops, which equals \(\dim \ker H_N\) by Corollary 12.1.79. The inclusion is consequently an equality.
Let \(B\) be normal and in left-canonical form. Suppose that the family \(A_j\) is a BNT for \(B\), and that the nearest-neighbor parent Hamiltonian of \(B\) has the all-chain ground-space property of Definition 12.1.17: for every \(N{\gt}2\), the length-two translated parent interactions commute, \(h_i(B,2)h_j(B,2)=h_j(B,2)h_i(B,2)\), the vector \(V^{(N)}(B)\) has zero energy, and \(\ker H_2^{(N)}(B) = \operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \). Then \(B\) is a renormalization fixed point.
Scope restriction (NNCPH representative): These two reverse-implication results cover only normal left-canonical and multiplicity-one unit-weight representatives, whereas [ 13 , Theorem 3.10 ] is stated for general canonical-form tensors with repeated copies and raw weights; documented in [ 167 ] .
The commuting interaction determines Beigi’s finite sector graph. Its positive loops span the parent ground space, and the normalized minimal loop tensors \(L_j\) are normal renormalization fixed points. For the normal representative \(B\),
Normal-tensor orthogonality applied to this eventual-span relation first identifies the bond dimensions and then gives an index \(j\), an invertible matrix \(X\), and a scalar \(\zeta \) with \(|\zeta |=1\) such that \(L_j^i = \zeta X B^i X^{-1}\). Transfer idempotence is preserved by these transformations.
Let \(A=(A_i)_i\) and \(B=(B_i)_i\) be two tensors, possibly with different bond dimensions, and let \(u\) be unitary on the physical index. If
then
Expand both physical sums. The coefficient of \(A_jX B_k^\dagger \) is \(\sum _i u_{ij}\overline{u_{ik}}=\delta _{jk}\), so only the diagonal terms remain.
Let \(U,A\) and \(V,B\) be pairs of tensors with respective common bond dimensions. Suppose that \(\zeta _A,\zeta _B\ne 0\) and that \(X_A,X_B\) are invertible matrices such that, for every physical index \(i\),
Then
Identifying equal bond dimensions on either tensor leg preserves this vanishing condition in both directions.
Set \(D_A=\zeta _A I\) and \(D_B=\zeta _B I\). For every rectangular matrix \(Z\), covariance of the mixed transfer map gives
If \(\mathcal E_{A,B}=0\), the left-hand side vanishes. The two outer factors are invertible, so cancellation gives \(\mathcal E_{U,V}(Z)=0\) for every \(Z\).
If \(a\) and \(b\) are distinct positive loops in the sector graph, then the mixed transfer map of their normalized minimal loop tensors vanishes:
In sector coordinates, every physical letter of \(L_a\) is supported in sector \(a\), while every physical letter of \(L_b\) is supported in sector \(b\). Hence, for every rectangular matrix \(Z\) and every physical index \(i\),
The common one-site physical unitary preserves the mixed transfer map by Lemma 12.1.92.
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and set
Suppose that, for every \(N{\gt}2\), the translated length-two parent interactions of \(B\) commute, every \(V^{(N)}(A_j)\) has zero energy, and
Then \(B\) is a renormalization fixed point. This establishes the reverse implication of [ 13 , Theorem 3.10 ] for multiplicity-one unit-weight representatives.
Beigi’s positive-loop product states span each parent ground space. Consequently every normal basis tensor \(A_j\) agrees, up to a unit-modulus scalar and an invertible virtual change of basis, with a normalized minimal loop tensor \(L_{\ell (j)}\). Distinctness of the BNT basis makes \(j\mapsto \ell (j)\) injective. Thus there are \(\zeta _j\ne 0\) and invertible \(X_j\) such that
Each \(L_{\ell (j)}\) has an idempotent transfer map. If \(j\ne k\), then \(L_{\ell (j)}\) and \(L_{\ell (k)}\) have disjoint sector support, so their mixed transfer map vanishes. Lemma 12.1.93 transports this vanishing through the two virtual changes of basis:
Thus the diagonal mixed transfer maps of the family \(A_j\) are idempotent and the off-diagonal ones vanish. The blockwise criterion is
for every pair \(j,k\), and proves the claim.
Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and set \(A=\bigoplus _{j=1}^g A_j\). Then
This statement concerns the multiplicity-one, unit-weight direct sum; it does not include repeated copies or arbitrary raw weights.
12.2 Decorrelation and the idempotent-product parent-commuting condition
In [ 13 , Appendix D.2 ] , the parent commuting Hamiltonian condition is stated for local projectors \(Q_{AX}\) and \(Q_{XB}\). With \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) denoting their lifts to the tripartite space, the commutation condition and common ground space \(K_{AXB}\) satisfy
Setting \(P_{AX}=1-Q_{AX}\) and \(P_{XB}=1-Q_{XB}\), the corresponding product identity on the tripartite space is \(\widehat P_{AX}\widehat P_{XB}=P_K\). For a positive semidefinite operator on a bipartite space, the support projector of the retained left marginal, lifted to the full space, fixes the original operator, as used in [ 13 , Appendix D.2 ] .
Scope restriction (abstract idempotent-product condition): The subsequent formal predicate retains only abstract commuting idempotents and their product identity; unlike [ 13 , Appendix D.2 ] , it omits tensor-product locality, orthogonal projectors, and the ground-space intersection, and admits the tautological choice \(P_{AX}=P_{XB}=P_K\). Documented in [ 139 ] .
For finite-dimensional spaces \(H_L\) and \(H_R\), one has \(\operatorname{tr}_R(\mathbb {1}_{L\otimes R})=(\dim H_R)\mathbb {1}_L\).
In product-basis coordinates,
For \(\rho \ge 0\) on \(H_L\otimes H_R\) with \(P_L\) the orthogonal projector onto the support of \(\operatorname{tr}_R\rho \), the identities \((P_L\otimes \mathbb {1}_R)\rho =\rho \) and \(\rho (P_L\otimes \mathbb {1}_R)=\rho \) provide the \(P_{AX}\) support-projector identities in [ 13 , Appendix D.2 ] upon setting \(H_L=H_A\otimes H_X\) and \(H_R=H_B\), as developed in [ 216 ] .
If \(\rho =\rho ^\dagger \), then \(\operatorname{tr}_L\rho =(\operatorname{tr}_L\rho )^\dagger \).
Entrywise Hermiticity gives
Similarly, for the support projector \(P_R\) of \(\operatorname{tr}_L\rho \), one has \((\mathbb {1}_L\otimes P_R)\rho =\rho \) and \(\rho (\mathbb {1}_L\otimes P_R)=\rho \).
We now use the explicit tripartite indexing \(H_A\otimes (H_X\otimes H_B)\). Reassociation identifies it with \((H_A\otimes H_X)\otimes H_B\) whenever the \(AX\) marginal is taken.
Writing \(\widehat M_R\) for the lift of an operator \(M_R\) supported on \(R\in \{ AX,XB,A,B\} \), the four lifts to \(H_A\otimes (H_X\otimes H_B)\) are
Identities (21) and (22) use the canonical identification \((H_A\otimes H_X)\otimes H_B\cong H_A\otimes (H_X\otimes H_B)\).
The lifts from \(AX\) and \(XB\) preserve multiplication,
and each of the four lifts from \(AX\), \(XB\), \(A\), and \(B\) preserves adjoints:
For the two-factor lifts,
Moreover,
and similarly,
with the tensor factors reassociated to the tripartite space.
Regions \(A\) and \(B\) are decorrelated relative to \(P\) if, for all Hermitian observables \(O_A=O_A^\dagger \) and \(O_B=O_B^\dagger \), one has \(P O_A(\mathbb {1}-P)O_B P=0\). This is [ 13 , Definition D.1, lines 2187–2191 ] .
The complex-linear extension requires \(P O_A(\mathbb {1}-P)O_B P=0\) for arbitrary complex matrices \(O_A\) and \(O_B\).
Observable decorrelation holds if and only if its complex-linear extension holds.
Every complex matrix has the decomposition
where \(H=H^\dagger \) and \(K=K^\dagger \). For analogous decompositions \(O=H+\mathrm{i}K\) and \(O'=H'+\mathrm{i}K'\),
Each summand vanishes by observable decorrelation applied to its two Hermitian components.
If \(P=P^\dagger \) and \(A\) and \(B\) are decorrelated, then, for arbitrary complex matrices \(O_A\) and \(O_B\), one has \(P O_B(\mathbb {1}-P)O_A P=0\).
Apply decorrelation to \(O_A^\dagger \) and \(O_B^\dagger \), and take adjoints.
The two reduced operators are \(\rho _{AX}=\operatorname{tr}_B P\) and \(\rho _{XB}=\operatorname{tr}_A P\).
For matrix units on \(A\) and \(B\),
For the \(XB\) marginal, evaluation in the product basis gives
Hence
The \(AX\) identity follows symmetrically.
If \(A\) and \(B\) are decorrelated, then \(\widehat\rho _{XB}(\mathbb {1}-P)\widehat\rho _{AX}=0\).
Substitute the matrix-unit expansions. Every summand has the form
where the bracketed factor vanishes by decorrelation.
If \(P=P^\dagger \) and \(A\) and \(B\) are decorrelated, then \(\widehat\rho _{AX}(\mathbb {1}-P)\widehat\rho _{XB}=0\).
Substitute the matrix-unit expansions. Every summand has the form
where the bracketed factor vanishes by reverse-order decorrelation.
For a Hermitian operator \(P\), let \(P_{AX}\) and \(P_{XB}\) be the support projectors of \(\operatorname{tr}_B P\) and \(\operatorname{tr}_A P\), respectively, as in [ 13 , Appendix D.2, lines 2225–2229 ] . Write \(\widehat P_{AX}\) and \(\widehat P_{XB}\) for their lifts to the tripartite space.
If \(P=P^\dagger \) and \(A\) and \(B\) are decorrelated, then
For \(g(0)=0\) and \(g(x)=x^{-1}\) when \(x\ne 0\), functional calculus gives
Hence
and, in the reverse order,
For \(P\geq 0\), \(\widehat P_{XB}P=P=P\widehat P_{XB}\).
For the bipartition \(A|XB\), the two support-absorption identities give
For \(P\geq 0\), \(\widehat P_{AX}P=P=P\widehat P_{AX}\).
After reassociating the tripartite space as \(AX|B\), the two support-absorption identities give
If \(P=P^\dagger =P^2\) and \(A\) and \(B\) are decorrelated relative to \(P\), then
Inserting \(\mathbb {1}=P+(\mathbb {1}-P)\) between the two support projectors gives
where \(\widehat P_{XB}P\widehat P_{AX}=P\) by the support-absorption identities and \(\widehat P_{XB}(\mathbb {1}-P)\widehat P_{AX}=0\) by (23). The same calculation with the two factors reversed, using (24), gives (26).
There are orthogonal projectors \(P_{AX}\) and \(P_{XB}\) on the two local factors whose lifts commute and whose ranges satisfy the source ground-space intersection condition \(\operatorname{ran}P=\operatorname{ran}\widehat P_{AX}\cap \operatorname{ran}\widehat P_{XB}\). Thus \(Q_{AX}=\mathbb {1}-P_{AX}\) and \(Q_{XB}=\mathbb {1}-P_{XB}\) are exactly the commuting parent terms of [ 13 , Definition D.2, lines 2205–2218 ] .
Let \(P\), \(P_{AX}\), and \(P_{XB}\) be orthogonal projectors, and write their lifts to the tripartite space as \(\widehat P_{AX}=P_{AX}\otimes \mathbb {1}_B\) and \(\widehat P_{XB}=\mathbb {1}_A\otimes P_{XB}\). Suppose that the lifted projectors commute, \(\widehat P_{AX}\widehat P_{XB} =\widehat P_{XB}\widehat P_{AX}\). Then
This is the passage from [ 13 , Definition D.2, lines 2205–2218 ] to the projector identity at lines 2279–2289.
A vector lies in the range of \(\widehat P_{AX}\widehat P_{XB}\) precisely when it is fixed by both commuting idempotents. Hence \(\operatorname{ran}(\widehat P_{AX}\widehat P_{XB}) =\operatorname{ran}\widehat P_{AX}\cap \operatorname{ran}\widehat P_{XB}\). The product is again an orthogonal projector. Orthogonal projectors with equal ranges are equal, giving the forward implication in (27); the reverse implication follows by substituting the product identity.
Starting from decorrelation, take the support projectors of \(\rho _{AX}\) and \(\rho _{XB}\). The preceding product theorem proves their commutation and identifies their product with \(P\).
Starting from commuting parent projectors, first use the ground-space intersection theorem to write \(P=\widehat P_{AX}\widehat P_{XB} =\widehat P_{XB}\widehat P_{AX}\). Locality gives \([O_A,\widehat P_{XB}]=[O_B,\widehat P_{AX}]=0\), while \(\widehat P_{XB}(\mathbb {1}-P)\widehat P_{AX}=0\). Therefore, using the two locality commutators,
Combine the complex-linear parent equivalence with the decomposition \(O=H+\mathrm{i}K\) from Theorem 12.2.7.
The structure used below records only the idempotent-product consequence of the parent commuting Hamiltonian condition of [ 13 , Appendix D.2 ] . It consists of commuting idempotent endomorphisms \(P_{AX}\) and \(P_{XB}\) such that, for the idempotent \(P_K\) associated to \(K\), \(P_{AX}\circ P_{XB}=P_K\). In the source, this is the equation obtained from local projectors \(Q_{AX}\) and \(Q_{XB}\) satisfying \([Q_{AX},Q_{XB}]=0\) and \(K_{AXB}=(K_{AX}\otimes H_B)\cap (H_A\otimes K_{XB})\).
If \(P_K\circ P_K=P_K\), then \(P_K\) has idempotent-product data.
Take \(P_{AX}=P_{XB}=P_K\). Their commutativity is immediate, and the product identity is exactly the assumed idempotence of \(P_K\).
If \(P\) and \(Q\) are commuting idempotents, then \(Q\circ (\mathbb {1}-P\circ Q)\circ P=0\).
The cancellation is the computation
using commutativity and idempotence.
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_K\circ P_K=P_K\).
Since \(P_{AX}\) and \(P_{XB}\) are idempotent and commute, their product \(P_{AX}\circ P_{XB}\) is idempotent. Substituting \(P_K=P_{AX}\circ P_{XB}\) gives the claim.
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_{AX}\circ P_K=P_K\).
Using \(P_K=P_{AX}\circ P_{XB}\) and the idempotence of \(P_{AX}\),
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_K\circ P_{XB}=P_K\).
Using \(P_K=P_{AX}\circ P_{XB}\) and the idempotence of \(P_{XB}\),
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_{XB}\circ P_K=P_K\).
By commutativity and the idempotence of \(P_{XB}\),
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_K\circ P_{AX}=P_K\).
By commutativity and the idempotence of \(P_{AX}\),
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_{XB}\circ P_{AX}=P_K\).
This follows from commutativity of \(P_{AX}\) and \(P_{XB}\) together with the identity \(P_{AX}\circ P_{XB}=P_K\).
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data and \(Q_{AX}=\mathbb {1}-P_{AX}\) and \(Q_{XB}=\mathbb {1}-P_{XB}\), then \(Q_{AX}\circ Q_{XB}=Q_{XB}\circ Q_{AX}\).
Expanding both products and using \(P_{AX}\circ P_{XB}=P_{XB}\circ P_{AX}\) shows that the complementary projectors commute.
The key identity is \(P_{XB}\circ (\mathbb {1}-P_{AX}\circ P_{XB})\circ P_{AX}=0\). Substituting \(P_K=P_{AX}\circ P_{XB}\) and using \(O_A\circ P_{XB}=P_{XB}\circ O_A\) and \(O_B\circ P_{AX}=P_{AX}\circ O_B\) gives
Apply Theorem 12.2.33 to the witnessing endomorphisms \(P_{AX}\) and \(P_{XB}\) supplied by the idempotent-product data.
For any endomorphisms \(P\) and \(Q\),
Expanding the left-hand side of (28) and cancelling gives the result.
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then
where \(Q_{AX}=\mathbb {1}-P_{AX}\) and \(Q_{XB}=\mathbb {1}-P_{XB}\) are the complementary projectors. This is the algebraic identity obtained from the complementary projectors in [ 13 , Appendix D, Section D.2 ] ; it is not the full source parent commuting Hamiltonian condition.
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then
This is the algebraic consequence corresponding to equation (D.2) from [ 13 ] : \(K_{AXB}=(K_{AX}\otimes H_B)\cap (H_A\otimes K_{XB})\). The source equation itself also carries the tensor-product local-subspace hypotheses.
(\(\Rightarrow \)) If \(P_Kv=v\), then
by left absorption, and similarly for \(P_{XB}\).
(\(\Leftarrow \)) If \(P_{AX}v=v\) and \(P_{XB}v=v\), then
This proves (30).
12.3 Spectral gap
A frustration-free Hamiltonian is gapped if its smallest nonzero eigenvalue, or the spectral gap above the ground-state space, is bounded away from zero uniformly in the chain length \(N\). For parent Hamiltonians of matrix product states, the martingale method of Fannes, Nachtergaele, and Werner [ 194 , 225 ] , as formulated by Cirac et al. [ 17 , Section IV.C ] and Kastoryano and Lucia [ 207 ] , reduces a uniform spectral gap to an anticommutator estimate for overlapping local terms. Principal angles between local ground spaces provide a geometric interpretation of this criterion, and norm-compression bounds for cyclic-window local terms yield uniform spectral gap lower bounds on periodic chains.
Denote by \(\mathcal G_{N,L}^{\mathrm{ES}}(A)\) the parent ground space after identifying the coefficient space with the Hilbert space \(\ell ^2(\{ 0,\ldots ,d-1\} ^N)\).
Denote by \(H_{N,L}^{\mathrm{ES}}(A)\) the conjugate of \(H_N(A,L)\) by the canonical identification between the coefficient space and \(\ell ^2(\{ 0,\ldots ,d-1\} ^N)\).
Under the canonical \(\ell ^2\) identification of the \(N\)-site state space, the parent ground space is exactly the kernel of the corresponding parent Hamiltonian.
This is an immediate unraveling of the definitions: both constructions are obtained from the same parent Hamiltonian by conjugating with the canonical identification between the site space and the \(\ell ^2\) space, so the ground space is precisely the kernel after this identification.
A vector belongs to \(\mathcal G_{N,L}^{\mathrm{ES}}(A)\) exactly when the parent Hamiltonian annihilates it.
Let \(P_L^{\mathrm{ES}}(A)\) denote the orthogonal projector onto \((G_L^{\mathrm{ES}}(A))^\perp \) on the canonical \(\ell ^2\) \(L\)-site Hilbert space. For a cyclic window starting at \(i\) and an outside configuration \(\tau \), the map \(R_{i,\tau }\) restricts an \(N\)-site vector to that length-\(L\) cyclic window with the outside configuration fixed by \(\tau \). The local term \(h_{i,\mathrm{ES}}\) is the corresponding \(N\)-site local projector, and \(R_{i,\tau }^{\dagger }P_L^{\mathrm{ES}}(A)R_{i,\tau }\) is the basic positive summand in its cyclic-averaging formula.
The Hilbert-space local projector \(P_L^{\mathrm{ES}}(A)\) is positive. Consequently, every summand \(R_{i,\tau }^{\dagger }P_L^{\mathrm{ES}}(A)R_{i,\tau }\) is positive, and so is the finite sum over \(\tau \).
Orthogonal projectors are positive. Conjugation by \(R_{i,\tau }\) preserves positivity, and finite sums of positive operators remain positive.
The Hilbert-space parent Hamiltonian is the sum of the Hilbert-space local terms, \(H_{N,\mathrm{ES}}=\sum _i h_{i,\mathrm{ES}}\), and each local term admits the exact cyclic-averaging formula
Unfold the Hilbert-space local term and compare it configuration-by-configuration with the cyclic restrictions. The \(d^{-L}\) factor compensates for the \(d^L\) choices of the window coordinates in the ambient configuration parameter \(\tau \).
Each Hilbert-space local term \(h_{i,\mathrm{ES}}\) is positive, hence the full Hilbert-space parent Hamiltonian \(H_{N,\mathrm{ES}}\) is positive.
Apply the averaging formula (31): \(h_{i,\mathrm{ES}}\) is a non-negative scalar multiple of a finite sum of positive conjugates of \(P_L^{\mathrm{ES}}(A)\), hence is itself positive. Summing over \(i\) yields positivity of \(H_{N,\mathrm{ES}}\).
The Hilbert-space local projector \(P_L^{\mathrm{ES}}(A)\) is a symmetric projection, and every Hilbert-space local term \(h_{i,\mathrm{ES}}\) on the \(N\)-site chain is a symmetric projection.
The local operator \(P_L^{\mathrm{ES}}(A)\) is an orthogonal projection onto \((G_L^{\mathrm{ES}}(A))^\perp \). For \(L\le N\), restricting \(h_{i,\mathrm{ES}}v\) to a cyclic window gives \(P_L^{\mathrm{ES}}(A)\) applied to the corresponding restriction of \(v\); applying \(h_{i,\mathrm{ES}}\) a second time therefore applies \((P_L^{\mathrm{ES}}(A))^2=P_L^{\mathrm{ES}}(A)\) on that window. Lemma 12.3.8 supplies symmetry, and the case \(L{\gt}N\) is the zero projection by definition.
The Hilbert-space local interaction \(P_L^{\mathrm{ES}}(A)\) annihilates exactly the Hilbert-space local ground space: \(P_L^{\mathrm{ES}}(A)v=0\) if and only if \(v\in G_L^{\mathrm{ES}}(A)\).
Since \(P_L^{\mathrm{ES}}(A)\) is the orthogonal projector onto \((G_L^{\mathrm{ES}}(A))^\perp \), its kernel is the original subspace \(G_L^{\mathrm{ES}}(A)\).
For \(L\le N\), a Hilbert-space local term \(h_{i,\mathrm{ES}}\) annihilates a vector \(v\) if and only if every restriction of \(v\) to the cyclic length-\(L\) window starting at \(i\), with the outside configuration fixed, lies in \(G_L^{\mathrm{ES}}(A)\).
Restricting \(h_{i,\mathrm{ES}}v\) to the same cyclic window gives \(P_L^{\mathrm{ES}}(A)\) applied to the corresponding restriction of \(v\). Lemma 12.3.10 identifies the vanishing of this local projector with membership in \(G_L^{\mathrm{ES}}(A)\).
Let \(\iota _L\) denote the canonical identification of the \(L\)-site state space with its Hilbert-space realization, and let \(R^{\mathrm{cyc}}_{i,\tau }\) denote cyclic restriction to the length-\(L\) window beginning at \(i\), with background configuration \(\tau \). For every \(L\in \mathbb {N}\), every \((L+1)\)-site Hilbert-space vector \(v\), and every background configuration \(\tau \),
On three sites, for every one-site prefix \(u\) and every state \(\psi \),
Finally, if \(L{\gt}0\), then for every \((L+1)\)-site Hilbert-space vector \(v\) and every background configuration \(\tau \),
No hypothesis on the physical or virtual dimension is required.
Expand the restriction maps on a configuration. At site \(0\), cyclic order is ordinary order and fixing the complement fixes the final coordinate. A one-site prefix is its unique coordinate, which gives the middle identity. If \(L{\gt}0\), cyclic restriction beginning at site \(1\) fixes the first coordinate and preserves the order of the remaining \(L\) coordinates.
For a tensor \(B\), let \(h_0\) and \(h_1\) be the two translated length-two parent interactions on the three-site Hilbert space. Then
Consequently, the orthogonal projectors onto the complementary excitation subspaces satisfy
These identities require neither injectivity nor a nonzero bond dimension.
At site \(0\), fixing the final physical coordinate is the cyclic restriction defining \(h_0\). At site \(1\), fixing the one-site prefix is the first-site restriction and hence the cyclic restriction defining \(h_1\). The kernel characterization of a translated local parent interaction therefore gives the two displayed kernel identities. Each \(h_i\) is an orthogonal projector, so its range is the orthogonal complement of its kernel. Uniqueness of the orthogonal projector with a prescribed range gives the final two identities.
Assume \(N{\gt}0\) and \(L\le N\). If a vector \(\psi \) has zero energy for the finite periodic parent Hamiltonian, \(H_N(A,L)\psi =0\), then every cyclic length-\(L\) restriction of \(\psi \) lies in the local MPS ground space. Equivalently, \(\ker H_N(A,L)\subseteq G_L^{(N)}(A)\).
Let \(e_N\) be the canonical \(\ell ^2\) identification from the Hilbert-space model to the coefficient space. Since \(H_N(A,L)\) is the \(e_N\)-conjugate of \(H_{N,L}^{\mathrm{ES}}(A)\), the hypothesis gives
Put \(P_i=h_{i,\mathrm{ES}}\) and \(H=\sum _iP_i\). If \(Hv=0\), then
Each summand is non-negative; hence \(\| P_i v\| ^2=0\) and therefore \(P_i v=0\) for every \(i\). Applying this to \(H_{N,L}^{\mathrm{ES}}(A)=\sum _i h_{i,\mathrm{ES}}\) gives the vanishing of every Hilbert-space local term. Lemma 12.3.11 identifies these equations with the conditions \(R_{i,\tau }(e_N^{-1}\psi )\in G_L^{\mathrm{ES}}(A)\) for all \(i\) and \(\tau \). Transporting these conditions back through \(e_N\) gives \(R_{i,\tau }\psi \in G_L(A)\) for all \(i\) and \(\tau \), which are exactly the cyclic-window defining equations of \(G_L^{(N)}(A)\).
Assume \(N{\gt}0\) and \(L\le N\). Then \(G_L^{(N)}(A)\subseteq \ker H_N(A,L)\).
If \(\psi \in G_L^{(N)}(A)\), then every local parent interaction annihilates \(\psi \), for every site \(i\): \(h_i(A,L)\psi =0\). Summing over all sites gives \(H_N(A,L)\psi =\sum _i h_i(A,L)\psi =0\).
The Hilbert-space cyclic constraint space is the transport of \(G_L^{(N)}(A)\) by the canonical \(\ell ^2\) identification: \(G_{L,\mathrm{ES}}^{(N)}(A)=e_N^{-1}G_L^{(N)}(A)\).
Assume \(N{\gt}0\) and \(L\le N\). Then \(\ker H_N(A,L)=G_L^{(N)}(A)\). Equivalently, after the canonical Hilbert-space identification, \(\mathcal K_N^{\mathrm{ES}}(A,L)=e_N^{-1}G_L^{(N)}(A)\), where \(\mathcal K_N^{\mathrm{ES}}(A,L)\) is the zero-energy space of \(H_{N,L}^{\mathrm{ES}}(A)\).
The inclusion \(\ker H_N(A,L)\subseteq G_L^{(N)}(A)\) is Theorem 12.3.14. For the converse, a vector in \(G_L^{(N)}(A)\) satisfies all local equations \(h_i(A,L)\psi =0\). Therefore \(H_N(A,L)\psi =\sum _i h_i(A,L)\psi =0\). The Hilbert-space statement is the same equality transported by \(e_N^{-1}\).
If \(A\) is injective and \(L\ge 2\), then on an \((L+1)\)-site chain the kernels of the two adjacent Hilbert-space \(L\)-site local terms have common kernel equal to the Hilbert-space \((L+1)\)-site MPS ground space:
Lemma 12.3.11 translates the two local kernel assumptions into the left and right \(L\)-site ground conditions. The open-chain intersection property, Theorem 11.5.4.3, then grows back the shared \((L-1)\)-site overlap to the \((L+1)\)-site ground space. The converse uses the forward restriction lemmas for ground-space vectors and the same kernel–restriction characterization.
Let \(L\le N\) and let \(h_{i,\mathrm{ES}},h_{j,\mathrm{ES}}\) be Hilbert-space local terms whose cyclic \(L\)-windows are site-disjoint: no site of the periodic chain has cyclic offset \({\lt}L\) from both \(i\) and \(j\). Then the terms commute pointwise: for every vector \(v\), \(h_{i,\mathrm{ES}}(h_{j,\mathrm{ES}}v) =h_{j,\mathrm{ES}}(h_{i,\mathrm{ES}}v)\). The same commutation identity holds for the corresponding local terms before passing to the Hilbert-space model. Their ordered cross term is non-negative: \(0\le \operatorname{Re}\langle h_{i,\mathrm{ES}}v,h_{j,\mathrm{ES}}v\rangle \).
In coordinates, \(h_{i,\mathrm{ES}}\) applies \(P_L^{\mathrm{ES}}(A)\) only to the \(i\)-window variables and leaves the complementary variables fixed. Site disjointness gives two replacement identities: replacing the \(i\)-window does not change the extracted \(j\)-window, and the \(i\)- and \(j\)-window replacements commute. Therefore the two embedded local operators commute. Remark 12.3.1.8 gives the non-negative ordered cross term for commuting symmetric projections.
Let \(H\) be a positive semidefinite self-adjoint operator on a finite-dimensional complex Hilbert space, and suppose that, for all \(v\), \(\gamma \langle v,Hv\rangle \le \langle Hv,Hv\rangle \), i.e., \(H^2\ge \gamma H\) as a quadratic form, for some \(\gamma {\gt}0\). Then, for every \(v\) in the orthogonal complement of \(\ker H\), \(\gamma \| v\| \le \| Hv\| \). In particular, every positive eigenvalue of \(H\) is at least \(\gamma \). The positive-semidefiniteness hypothesis is essential: without it, \(H=-\mathbb {1}\) with \(\gamma =2\) satisfies the quadratic-form inequality vacuously but violates the norm bound.
By the finite-dimensional spectral theorem, \(H\) diagonalises in an orthonormal eigenbasis with real eigenvalues \(\lambda _k\). For a unit eigenvector \(\psi _k\) with eigenvalue \(\lambda _k\), the hypothesis gives \(\gamma \lambda _k\le \lambda _k^2\), whence \(\lambda _k\le 0\) or \(\lambda _k\ge \gamma \). Since \(H\) is positive semidefinite (its quadratic form is \(\ge 0\)), the negative case is excluded and every nonzero eigenvalue satisfies \(\lambda _k\ge \gamma \). Writing \(v=\sum _k c_k\psi _k\) and projecting onto \((\ker H)^\perp \) keeps only terms with \(\lambda _k\ge \gamma \), so
12.3.1 Martingale condition for a finite family of projections
Let \(H_{N,L}^{\mathrm{ES}}(A)\) be the Hilbert-space representative of the parent Hamiltonian. If a constant \(\gamma {\gt}0\) satisfies the uniform quadratic-form estimate
for every \(N\ge 2L\) and every vector \(v\), then \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \) for every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \). In particular, with \(L{\gt}1\) and \(\gamma =1/(4L)\), the explicit gap-bound conclusion follows once this same quadratic-form estimate (37) is supplied.
Let \(U\) and \(V\) be subspaces of a finite-dimensional complex Hilbert space. Their Friedrichs overlap is at most \(\eta \) when
for every \(x\in U\cap (U\cap V)^\perp \) and \(y\in V\cap (U\cap V)^\perp \).
Let \(P_U\), \(P_V\), and \(P_{U\cap V}\) be the orthogonal projections onto \(U\), \(V\), and their intersection. If
then the Friedrichs overlap of \(U\) and \(V\) is at most \(\eta \).
For \(y\in V\cap (U\cap V)^\perp \), the defect applied to \(y\) is \(P_Uy\). Hence \(\| P_Uy\| \le \eta \| y\| \). If also \(x\in U\cap (U\cap V)^\perp \), self-adjointness of \(P_U\) gives \(\langle x,y\rangle =\langle x,P_Uy\rangle \), and Cauchy–Schwarz proves the claim.
Let \(P_U\) be the orthogonal projection onto \(U\). If the Friedrichs overlap of \(U\) and \(V\) is at most \(\eta \), where \(\eta \ge 0\), then
for every \(v\in V\cap (U\cap V)^\perp \).
Put \(x=P_Uv\). Since the common intersection is contained in \(U\), the projection preserves orthogonality to \(U\cap V\). Thus \(x\in U\cap (U\cap V)^\perp \), and \(\langle x,v\rangle =\langle x,x\rangle =\| x\| ^2\). The Friedrichs bound gives \(\| x\| ^2\le \eta \| x\| \| v\| \), and cancellation proves the result when \(x\ne 0\); the case \(x=0\) is immediate.
Let \(P_U\) and \(P_V\) be the orthogonal projections onto \(U\) and \(V\). If the Friedrichs overlap of \(U\) and \(V\) is at most \(\eta \), where \(\eta \ge 0\), then every \(v\in (U\cap V)^\perp \) satisfies
The vector \(P_Vv\) belongs to \(V\) and remains orthogonal to \(U\cap V\). Apply Lemma 12.3.1.4.
Under the hypotheses of Lemma 12.3.1.5, every \(v\in U\cap (U\cap V)^\perp \) satisfies
Orthogonal projection onto \(V\) does not increase the norm, while \(P_Uv=v\). Hence \(\| P_UP_Vv\| \le \eta \| P_Vv\| \le \eta \| v\| =\eta \| P_Uv\| \).
Let \(P\) and \(Q\) be the orthogonal projections onto \(K^\perp \) and \(L^\perp \). If the Friedrichs overlap of these two ranges is at most \(\eta \ge 0\), then every \(v\in K^\perp \cap (K^\perp \cap L^\perp )^\perp \) satisfies
Apply Lemma 12.3.1.6 to the two subspaces \(K^\perp \) and \(L^\perp \).
The projection-geometry reduction uses the following elementary Hilbert-space identities: the diagonal identity \(\operatorname{Re}\langle Pv,Pv\rangle =\operatorname{Re}\langle Pv,v\rangle \) for symmetric projections; nonnegativity of this quadratic form; nonnegativity of \(\operatorname{Re}\langle Pv,Qv\rangle \) for commuting symmetric projections \(P,Q\); the Cauchy–Schwarz conversion from \(\| PQv\| \le c\| Pv\| \) to \(\operatorname{Re}\langle Pv,Qv\rangle \ge -c\operatorname{Re}\langle Pv,v\rangle \); the symmetric variant which, for \(c\ge 0\), converts the operator-product bound \(\| PQv\| \le c\| Qv\| \) to the anticommutator estimate \(\operatorname{Re}\langle Pv,Qv\rangle +\operatorname{Re}\langle Qv,Pv\rangle \ge -c(\operatorname{Re}\langle Pv,v\rangle +\operatorname{Re}\langle Qv,v\rangle )\); finite-sum expansions of \(\operatorname{Re}\langle (\sum _iP_i)v,v\rangle \) and \(\operatorname{Re}\langle (\sum _iP_i)v,(\sum _iP_i)v\rangle \); the conclusion that a vector annihilated by a finite sum of symmetric projections is annihilated by each projection; the diagonal/off-diagonal split of the ordered double sum; the aggregate off-diagonal estimate obtained from row sums \(\sum _{j\ne i}c_{ij}\le 1\); and the corresponding aggregate estimate for the anticommutator martingale condition.
Let \(P_i\) be a finite family of symmetric projections and let \(H=\sum _iP_i\). If the ordered off-diagonal forms obey
with row sums \(\sum _{j\ne i}c_{ij}\le 1\) and \(\gamma \le 1\), then \(H^2\ge \gamma H\) as a quadratic form.
Expand \(\operatorname{Re}\langle Hv,Hv\rangle \) as an ordered double sum. Symmetric idempotence identifies each diagonal term with \(\operatorname{Re}\langle P_i v,v\rangle \), while the row-summable ordered cross-term bounds control the off-diagonal contribution by \(-(1-\gamma )\sum _i\operatorname{Re}\langle P_i v,v\rangle \).
Let \(P_i\) be a finite family of symmetric projections and let \(H=\sum _iP_i\). Suppose the anticommutator forms obey
with row sums \(\sum _{j\ne i}c_{ij}\le 1\), column sums \(\sum _{i\ne j}c_{ij}\le 1\), and \(\gamma \le 1\). Then \(H^2\ge \gamma H\) as a quadratic form.
Expand \(\operatorname{Re}\langle Hv,Hv\rangle \) as an ordered double sum. The anticommutator bound controls the sum of each ordered term with its reversed term. The row and column estimates bound the coefficient-weighted diagonal contribution by twice \(\sum _i\operatorname{Re}\langle P_i v,v\rangle \); after dividing by two, the same quadratic-form estimate follows.
Let \(P_i\) be a finite family of symmetric projections and let \(\gamma \le 1\). Suppose an interaction predicate \(i\sim j\) has at most \(m\) off-diagonal neighbours in each row and in each column, where \(m{\gt}0\). If noninteracting pairs have non-negative anticommutator form and interacting pairs satisfy
then \(H=\sum _iP_i\) satisfies \(H^2\ge \gamma H\) as a quadratic form.
Choose \(c_{ij}=1/m\) on interacting pairs and \(c_{ij}=0\) otherwise. The row and column cardinality bounds give the corresponding coefficient sums, and the noninteracting anticommutator non-negativity supplies the zero-coefficient estimates. Apply Lemma 12.3.1.10.
Let \(P_i\) be a finite family of symmetric projections and let \(\gamma \le 1\). Suppose an interaction predicate \(i\sim j\) has at most \(m\) off-diagonal neighbours in each row for some positive integer \(m\). If noninteracting ordered cross terms are non-negative and, on each interacting pair,
then \(H=\sum _iP_i\) satisfies \(H^2\ge \gamma H\) as a quadratic form.
Choose \(c_{ij}=1/m\) on interacting pairs and \(c_{ij}=0\) otherwise. The row cardinality bound gives \(\sum _{j\ne i}c_{ij}\le 1\); noninteracting nonnegativity supplies the zero-coefficient cross-term estimates. Apply Lemma 12.3.1.9.
Let \(P_i\) be a finite family of symmetric projections and let \(\gamma \le 1\). Suppose an interaction predicate has at most \(m{\gt}0\) off-diagonal neighbours in each row, noninteracting ordered cross terms are non-negative, and interacting pairs satisfy the norm-compression estimate
Then \(H=\sum _iP_i\) satisfies \(H^2\ge \gamma H\) as a quadratic form. The same conclusion holds from a separate compression coefficient \(\eta \) whenever \(\eta \le (1-\gamma )/m\). In particular, if \(0\le \eta \) and \(\eta m{\lt}1\), then \(H^2\ge (1-\eta m)H\) and \(1-\eta m{\gt}0\).
Cauchy–Schwarz and symmetric idempotence turn the norm-compression estimate into \(\operatorname{Re}\langle P_i v,P_jv\rangle \ge -(1-\gamma )m^{-1}\operatorname{Re}\langle P_iv,v\rangle \). If a separate coefficient \(\eta \) is used, the assumption \(\eta \le (1-\gamma )/m\) first weakens that estimate to (38). The finite-overlap row-sum reduction then applies. For the strict form, take \(\gamma =1-\eta m\).
If \(L\leq N\) and \(i\in \{ 0,\ldots ,N{-}1\} \), identify the sites of the periodic chain with the disjoint union of the cyclic window \(i,i+1,\ldots ,i+L-1\) and its ordered complement \(i+L,i+L+1,\ldots ,i+N-1\), with all indices taken modulo \(N\).
Let \(M\) be a commutative monoid. If \(L\leq N\), \(i\in \{ 0,\ldots ,N{-}1\} \), and \(f\colon \{ 0,\ldots ,N{-}1\} \to M\), then the product of \(f\) over the chain equals the product over the cyclic window beginning at \(i\) times the product over its ordered complement.
Reindex the product by the cyclic window and complement identification of Definition 12.3.1.14, then split the product over the resulting disjoint union.
Fix a tensor \(A\) with physical dimension \(d{\gt}0\) and a coefficient \(c\in \mathbb R\). For every chain length \(N\) and site \(i\in \{ 0,\ldots ,N{-}1\} \), set \(h_i^{(N)}:=h_{i,\mathrm{ES}}^{(N,2)}(A)\), the Euclidean range-two parent term of this same tensor \(A\) at \(i\). Suppose that, for every three-site vector \(w\),
Then, for every \(N\geq 3\), every site \(i\in \{ 0,\ldots ,N{-}1\} \), and every \(N\)-site vector \(v\), the forward pair satisfies
and the backward pair satisfies
All site indices are taken modulo \(N\), so these conclusions include pairs and three-site windows crossing the periodic cut.
For a cyclic three-site window beginning at \(i\), write \(\mathcal C_{N-3}\) for its ordered complementary configurations and \(R_{i,\tau }^{(3)}\) for restriction with complement \(\tau \) fixed. Definition 12.3.1.14 identifies every chain configuration uniquely with its window and complement, while the cyclic restriction operators of Remark 11.5.5.2 give the fiber decomposition
The restriction identities in Theorem 12.1.58 intertwine the two adjacent terms:
Applying (39) to \(w_\tau =R_{i,\tau }^{(3)}v\) and summing over the complement gives
By the fiber decomposition (42) and restriction intertwining (43), this is (40); in particular, summation leaves \(c\) unchanged. For the backward pair, begin the forward window at \(i-1\). Moving one step forward from that site returns \(i\), including when \(i=0\), and yields (41).
For a fixed chain length \(N\), suppose the Hilbert-space local terms \(h_{i,\mathrm{ES}}\) satisfy the ordered row-sum estimate
with \(\sum _{j\ne i}c_{ij}\le 1\) and \(\gamma \le 1\). Then the Hilbert-space parent Hamiltonian satisfies
for every vector \(v\).
Assume uniformly for every \(N\ge 2L\) that the Hilbert-space local terms satisfy the ordered row-sum estimate (45) with \(\gamma =1/(4L)\). If \(L{\gt}1\), then \(1/(4L){\gt}0\) and, for every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\frac{1}{4L}\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).
The fixed-chain row-sum reduction supplies the quadratic-form estimate with \(\gamma =1/(4L)\) for every admissible \(N\). The positivity and kernel identification of the Hilbert-space Hamiltonian then allow Lemma 12.3.1.1 to convert this quadratic-form estimate into the norm lower bound on the orthogonal complement of the ground space.
Assume uniformly for every \(N\ge 2L\) that the Hilbert-space local terms satisfy the anticommutator martingale estimate
for \(i\ne j\), with row and column coefficient sums bounded by one. If \(L{\gt}1\), then \(1/(4L){\gt}0\) and, for every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\frac{1}{4L}\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).
The fixed-chain anticommutator row-sum reduction supplies the quadratic-form estimate with \(\gamma =1/(4L)\) for every admissible \(N\). The positivity and kernel identification of the Hilbert-space Hamiltonian then convert this quadratic-form estimate into the norm lower bound on the orthogonal complement of the ground space.
Fix \(L{\gt}1\) and set \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) that the overlap relation on local windows has at most \(m\) off-diagonal neighbours in each row and in each column. If non-overlapping local terms have non-negative anticommutator form, and overlapping local terms satisfy
then the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
Apply the finite-overlap anticommutator projection reduction to the Hilbert-space local terms. The resulting quadratic-form estimate has \(\gamma =1/(4L)\), and Lemma 12.3.1.1 converts it to the norm lower bound.
For a fixed chain length \(N\), let \(m\) be a positive integer and let \(\gamma \le 1\). Suppose an overlap predicate on local windows has at most \(m\) overlapping off-diagonal windows in each row. If non-overlapping local terms have non-negative ordered cross terms, and overlapping local terms satisfy the one-sided ordered cross-term estimate
and the Hilbert-space local terms are symmetric projections, then the Hilbert-space Hamiltonian satisfies \(H_{N,L}^{\mathrm{ES}}{}^2\ge \gamma H_{N,L}^{\mathrm{ES}}\) as a quadratic form.
Apply the finite-overlap projection reduction with \(P_i=h_{i,\mathrm{ES}}\).
12.3.2 Cyclic-window overlap estimates
For a length-\(L\) cyclic window on \(\{ 0,\ldots ,N{-}1\} \) starting at \(i\), its support is \(S_i=\{ i,i+1,\ldots ,i+L-1\} \pmod N\), with repeated visits counted once.
If \(L\le N\), then a site \(k\) belongs to the cyclic support \(S_i\) exactly when its cyclic offset from \(i\) is below \(L\): \(k\in S_i \quad \Longleftrightarrow \quad (k-i)\bmod N {\lt} L\).
Write \(k=i+r\pmod N\) for a representative offset \(r\). Membership in \(S_i\) gives an offset \(r{\lt}L\), and conversely an offset below \(L\) gives the corresponding point of the support.
Two length-\(L\) cyclic windows overlap, written \(i\sim _L j\), when their supports meet: \(S_i\cap S_j\ne \varnothing \).
For length-\(L\) cyclic windows, \(i\sim _Lj\) if and only if \(j\sim _Li\).
Both assertions say that the two cyclic supports have a common site.
If \(L\le N\) and the cyclic supports \(S_i\) and \(S_j\) do not meet, then the Hilbert-space local terms have non-negative ordered cross term: \(0\le \operatorname{Re}\langle h_{i,\mathrm{ES}}v,h_{j,\mathrm{ES}}v\rangle \).
The support-offset characterization converts failure of \(S_i\cap S_j\ne \varnothing \) into site-disjointness of the two cyclic windows. The disjoint-window positivity lemma then applies.
If \(2L\le N\) and \(L{\gt}1\), then, for every \(i\in \{ 0,\ldots ,N{-}1\} \), \(\# \{ j\ne i: i\sim _L j\} \le 2(L-1)\).
If \(S_i\) and \(S_j\) meet, choose offsets \(a,b{\lt}L\) with \(i+a\equiv j+b\pmod N\). Since \(2L\le N\), the offset from \(i\) to \(j\) is either the clockwise distance \(a-b\in \{ 1,\ldots ,L-1\} \) or the counterclockwise distance \(b-a\in \{ 1,\ldots ,L-1\} \); the zero distance is excluded by \(j\ne i\). Thus every overlapping off-diagonal start lies among the \(L-1\) clockwise starts or the \(L-1\) counterclockwise starts.
Fix \(L{\gt}1\) and set \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) that the hypotheses of the fixed-chain finite-overlap martingale reduction hold with \(\gamma =1/(4L)\) and this value of \(m\). Then \(1/(4L){\gt}0\), and the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
The fixed-chain finite-overlap reduction gives the required quadratic-form estimate for every admissible \(N\). Lemma 12.3.1.1 converts that estimate to the norm lower bound.
Fix \(L{\gt}1\) and let \(i\sim _L j\) mean that the cyclic supports of the length-\(L\) windows at \(i\) and \(j\) meet. Suppose uniformly for every \(N\ge 2L\) that overlapping off-diagonal terms satisfy
Then \(1/(4L){\gt}0\), and the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
Lemma 12.3.2.6 gives the row-cardinality bound \(m=2(L-1)\) for the overlap relation among length-\(L\) local windows, and Lemma 12.3.9 supplies the symmetric projection structure of each Hilbert-space local term. If two cyclic supports do not meet, the corresponding windows are site-disjoint, so Lemma 12.3.2.5 gives the required non-negative ordered cross term. Apply Lemma 12.3.2.7 with these facts and the overlapping-window estimate (46).
Fix \(L{\gt}1\) and let \(i\sim _L j\) mean that the cyclic supports of the length-\(L\) windows at \(i\) and \(j\) meet. Suppose uniformly for every \(N\ge 2L\) that overlapping off-diagonal terms satisfy
Then the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
12.3.3 The weighted virtual matrix space
Let \(\rho \in M_D(\mathbb {C})\) be positive definite. The \(\rho \)-weighted inner product on \(M_D(\mathbb {C})\) is
This is the auxiliary inner product in [ 194 , Equation (5.6) ] .
For every \(X\in M_D(\mathbb {C})\),
By Theorem 12.3.3.1 with \(Y=X\),
The map
is a complex linear isometric isomorphism. Its inverse is
Thus the weighted matrix space has complex Hilbert dimension \(D^2\). This Euclidean realization is not asserted as a result of Fannes, Nachtergaele, and Werner.
12.3.4 Nested ground projections and martingale differences
Let \(\mathcal H\) be a complex Hilbert space. A nested ground-projection family is a sequence \((G_n)_{n\geq 0}\) of orthogonal projections such that
Its \(n\)th martingale difference is
Each \(E_n\) is an orthogonal projection, and
With Nachtergaele’s endpoint conventions \(G_0=\mathbf1\) and \(G_{N+1}=0\), the differences form a complete family:
These are the assertions following the display labelled \(E_n\) in [ 225 ] .
If \(m\leq n\), nestedness and orthogonality give \(G_mG_n=G_nG_m=G_n\). Consequently,
Self-adjointness follows from \(E_n^\dagger =G_n^\dagger -G_{n+1}^\dagger =E_n\). The complete sum telescopes with both endpoints visible:
Suppose \(G_0=\mathbf1\). For every \(v\perp \operatorname {ran}(G_N)\),
This is the identity resolutionpsi in the proof of [ 225 , Theorem 2.1(i) ] .
Since \(G_N\) is the orthogonal projection onto its range, \(v\perp \operatorname {ran}(G_N)\) implies \(G_Nv=0\). Hence
For every \(v\in \mathcal H\) and finite \(S\subset \mathbb {N}\),
In particular, for \(S=\{ 0,\ldots ,N-1\} \),
The summands are pairwise orthogonal because \(\langle E_mv,E_nv\rangle =\langle v,E_mE_nv\rangle =0\) for \(m\ne n\). The identity follows by the finite Pythagorean theorem.
Suppose \(G_0=\mathbf1\). For every \(v\perp \operatorname {ran}(G_N)\),
This is the identity resolutionnormpsi in the proof of [ 225 , Theorem 2.1(i) ] .
Substitute the truncated reconstruction \(v=\sum _{n=0}^{N-1}E_nv\) into the Pythagorean identity.
Let \(Q_n\) be a local ground projection and suppose that
Then every index outside \([n-l,n]\) cancels:
If \(n{\lt}N\), the complete resolution term therefore reduces to
This is the reduction from Enpsi to the first line of Enpsi2 in [ 225 , Theorem 2.1(i) ] .
For an outside index, commutation and orthogonality give
since either inequality implies \(m\ne n\). Hence only \(n-l\leq m\leq n\) remains in the finite sum. Finally, symmetry of each \(E_m\) gives
and summing over the active interval proves the displayed identity.
Fix an \(N\)-site Hilbert space and let \(h_i\) denote the nonwrapping interaction of length \(L\) starting at \(i\). For every \(m\geq 0\), set
All operators act on the same \(N\)-site space. The local interval ground projection in the proof of Theorem 2.1(i) is
The prefix and interval Hamiltonians are positive and symmetric:
Each \(h_i\) is an orthogonal projection and is therefore positive. Finite sums of positive operators are positive, and every positive operator is symmetric.
If \(m\leq n\), then
Hence \((G_m^{(N)})_{m\geq 0}\) is a nested ground-projection family.
If \(H_n^{(N)}v=0\), then orthogonality of the local projections gives
Every summand is nonnegative, so \(h_iv=0\) whenever \(i+L\leq n\). When \(m\leq n\), every summand of \(H_m^{(N)}\) occurs in \(H_n^{(N)}\), and hence \(H_m^{(N)}v=0\). This proves the kernel inclusion, and the range inclusion follows from \(\operatorname {ran}G_m^{(N)}=\ker H_m^{(N)}\).
Assume \(L\leq N\) and let \(E_m=G_m^{(N)}-G_{m+1}^{(N)}\). Then
Consequently, for \(n{\lt}N\) and every vector \(v\),
If \(m{\lt}n-l\), the supports of every summand of \(H_m^{(N)}\) and every summand of \(H_{[n-l,n+1)}\) are disjoint. Their local interactions commute, hence the two symmetric Hamiltonians commute. More generally, suppose that \(T\) is symmetric and that an arbitrary linear operator \(S\) commutes with \(T\). Then \(\ker S\) is invariant under \(T\), and symmetry of \(T\) makes \((\ker S)^\perp \) invariant as well. Thus \(P_{\ker S}\) commutes with \(T\). Since \(P_{\ker S}\) is symmetric, applying the same argument to \(\ker T\) shows
This proves the first range of indices for both \(G_m^{(N)}\) and \(G_{m+1}^{(N)}\).
If \(n{\lt}m\), every local interaction in \(H_{[n-l,n+1)}\) also occurs in \(H_m^{(N)}\). Therefore
and the corresponding nested orthogonal projections commute. Subtracting the two prefix identities gives \(E_mQ_n=Q_nE_m\) in both index ranges. The abstract outside-window cancellation theorem now restricts the full sum to \(m\in [n-l,n]\).
Let \(U:E\to F\) be a surjective complex linear isometry of inner product spaces, and let \(P:E\to E\) and \(Q:F\to F\) be linear maps satisfying \(Q=UPU^{-1}\). Then \(UPx=QUx\), \(\ker Q=U(\ker P)\) and \((\ker Q)^\perp =U((\ker P)^\perp )\). For every real \(\delta \), the estimate \(\delta \| x\| \leq \| Px\| \) on \((\ker P)^\perp \) is equivalent to \(\delta \| y\| \leq \| Qy\| \) on \((\ker Q)^\perp \). Finite dimensionality and positivity are not required.
Substituting \(Ux\) in the conjugacy gives \(UPx=QUx\). Injectivity of \(U\) identifies kernel membership, and surjectivity gives the kernel equality. The equality \(\langle Uz,Ux\rangle =\langle z,x\rangle \) identifies the orthogonal complements. Apply the respective norm estimate to \(x\) and \(Ux\) and use \(\| Ux\| =\| x\| \).
Let \(T\) and \(U\) be operators on a finite-dimensional active Hilbert space, and let \(S\) be a finite set of right-spectator configurations. The operator \(T^{\oplus S}\), defined by applying \(T\) independently at every \(s\in S\), satisfies
and
with equality in (80) whenever \(S\) is nonempty.
The squared norm splits as a sum over spectator fibers:
For nonempty \(S\), a vector supported on one spectator fiber gives the reverse inequality between the two operator norms. Addition, subtraction, and composition are checked independently on each fiber.
For \(n,r\in \mathbb {N}\), restriction to the first \(n\) sites and the final \(r\) sites gives a canonical equivalence
and the corresponding Euclidean linear isometry. A vector belongs to \(\ker (T^{\oplus S})\) exactly when each spectator fiber belongs to \(\ker T\). If \(P_{\ker T}\) is the orthogonal projection onto \(\ker T\), then
The configuration equivalence is the inverse of concatenation. The kernel characterization follows by evaluating each coordinate. Fiberwise extension preserves symmetry and idempotence, and its range consists exactly of the vectors whose fibers lie in \(\ker T\). Equality of the two symmetric projections follows from equality of their ranges.
Let \(i+L\leq n\). Under the canonical identification
the nonwrapping length-\(L\) interaction beginning at \(i\) is
The nonwrapping hypothesis identifies cyclic restriction with contiguous restriction on both chain lengths. Concatenation leaves the first \(n\) coordinates unchanged and fixes the final \(r\) spectator coordinates. Hence window extraction and restriction agree on every spectator fiber, and applying the same local parent interaction gives the identity.
Let \(L{\gt}0\) and \(p\leq n\). Under the canonical identification
both the prefix Hamiltonian on \([0,p)\) and every suffix-interval Hamiltonian ending at \(p\) are direct sums over the right-spectator configurations of their operators in the \(n\)-site volume.
The condition \(L{\gt}0\) makes the active starts in the two finite sums exactly the starts before \(p\). Concatenation gives a bijection between the corresponding start sets in the two volumes. Apply Theorem 12.3.4.14 to each summand and then sum over this bijection.
Let \(K,l,n,r\in \mathbb {N}\). Suppose that \(A\) has nonzero bond dimension, is block injective at length \(L_0{\gt}0\), and satisfies \(L_0+1\leq n\). The kernel of the range-\((L_0+1)\) prefix Hamiltonian on the first \(n\) sites of the \((n+1)\)-site volume is the left open-chain ground space. When that prefix instead fills its own \(n\)-site volume, its kernel is the whole \(n\)-site ground space. Independently, the kernel of the full final length-\((l+1)\) suffix window beginning at \(K\) is the tail open-chain ground space. Let \(U_{\mathrm{pre}}\) and \(U_{\mathrm{tail}}\) be the canonical right-spectator isometries for the active-volume decompositions \((n+1+r)=(n+1)+r\) and \((K+l+1+r)=(K+l+1)+r\), respectively. Their orthogonal projections satisfy
The prefix kernel is read fiberwise. Block injectivity identifies every fiber kernel with the \(n\)-site MPS ground space, which is precisely the left-window membership condition after fixing the last physical index. When the active prefix fills its smaller volume, the same argument identifies its kernel projection with the whole open-chain ground projection. The full suffix interval contains one interaction; its cyclic restriction is the ordinary restriction obtained by fixing the \(K\)-site prefix, giving the tail-window membership condition. Orthogonal kernel projections are preserved by the canonical isometry, and the fiberwise kernel-projection identity gives the two displayed formulas.
Suppose that \(0{\lt}l\) and \(l+1\leq N\). Take interaction length \(L=l+1\) and define \(E_n=G_n^{(N)}-G_{n+1}^{(N)}\). Then the two boundary ranges in the fixed \(N\)-site space are explicit:
At the remaining boundary index,
At \(n=l\), both \(Q_l\) and \(G_{l+1}^{(N)}\) are the kernel projection of the unique interaction on \([0,l+1)\). Thus
Hence
If \(n{\lt}l\), neither the \(n\)-site prefix nor the \((n+1)\)-site prefix contains a full length-\((l+1)\) interaction, so both ground projections are the identity, proving (89). At \(n=l\), the next prefix and the interval \([0,l+1)\) contain exactly the same single interaction. Idempotence of its kernel projection gives (90) via (91).
12.3.5 Spectator-indexed boundary maps
The tail and left boundary maps embed virtual-space data into a common \((K+L)\)-site Hilbert space, expressing overlapping ground spaces as ranges of linear maps on the same ambient space to compare their orthogonal projectors.
For \(K,L\in \mathbb {N}\) and a fixed suffix configuration \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \), the prefix restriction is the \(\mathbb {C}\)-linear map \(P_\tau \colon \mathbb {C}^{\{ 0,\ldots ,d{-}1\} ^{K+L}}\to \mathbb {C}^{\{ 0,\ldots ,d{-}1\} ^K}\) defined by \((P_\tau \psi )(\sigma )=\psi (\sigma ,\tau )\) for \(\sigma \colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \). This is the left-sided analogue of the suffix restriction of Definition 11.5.5.5.
For every boundary matrix \(X\in M_{D}(\mathbb {C})\) and every suffix \(\tau \),
Expand \(\Gamma _{K+L}(X)(\sigma ,\tau )=\operatorname{tr}(A^{\sigma ,\tau }X)\), factor \(A^{\sigma ,\tau }=A^\sigma A^\tau \) by Lemma 2.1.4, and reassociate: \(\operatorname{tr}((A^\sigma A^\tau )X)=\operatorname{tr}(A^\sigma (A^\tau X)) =\Gamma _K(A^\tau X)(\sigma )\).
For \(K,L\in \mathbb {N}\), the tail boundary map is the \(\mathbb {C}\)-linear map
defined for every family \(Y\colon (\{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} )\to M_{D}(\mathbb {C})\) by
where \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the prefix and \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the suffix. When \(Y_u=X A^u\) the output equals the standard \(\Gamma _{K+L}(X)\).
For \(K,L\in \mathbb {N}\), the left boundary map is the \(\mathbb {C}\)-linear map
defined for every family \(Z\colon (\{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} )\to M_{D}(\mathbb {C})\) by
where \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the prefix and \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the suffix. When \(Z_\tau =A^\tau X\) the output equals the standard \(\Gamma _{K+L}(X)\).
For every prefix \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) and suffix \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \),
Unfold the definition of \(\Gamma ^{\mathrm{tail}}_{K,L}\) on the pair \((u,\tau )=\{ 0,\ldots ,.{-}1\} \! append\; u\; \tau \), use the decomposition \((\{ 0,\ldots ,.{-}1\} \! append\; u\; \tau )\circ \{ 0,\ldots ,.{-}1\} \! natAdd\; K=\tau \) and \((\{ 0,\ldots ,.{-}1\} \! append\; u\; \tau )\circ \{ 0,\ldots ,.{-}1\} \! castAdd\; L=u\), and apply \(\Gamma _L(Y_u)(\tau )=\operatorname{tr}(A^\tau Y_u)\).
Restricting the tail boundary map to the suffix for a fixed prefix \(u\in \{ 0,\ldots ,d{-}1\} ^K\) recovers \(\Gamma _L(Y_u)\):
This is Theorem 12.3.5.5 applied pointwise: for each \(\tau \), \((R^{\mathrm{tail}}_u\psi )(\tau )=\psi (u,\tau )\).
The standard length-\((K+L)\) boundary map factors through the tail boundary map:
for every \(X\in M_{D}(\mathbb {C})\).
Write a configuration \(\sigma \in \{ 0,\ldots ,d{-}1\} ^{K+L}\) as \(\sigma =(u,\tau )\) with \(u\in \{ 0,\ldots ,d{-}1\} ^K\) and \(\tau \in \{ 0,\ldots ,d{-}1\} ^L\). Substitute \(Y_u=X A^u\) into the definition of \(\Gamma ^{\mathrm{tail}}_{K,L}\), apply the three-factor trace cycle \(\operatorname{tr}(A^\tau (X A^u))=\operatorname{tr}(A^u(A^\tau X))\), reassociate \(\operatorname{tr}(A^u(A^\tau X))=\operatorname{tr}((A^u A^\tau )X)\), factor \(A^u A^\tau =A^{u,\tau }\) by multiplicativity of word evaluation (Lemma 2.1.4), and identify the result with \(\Gamma _{K+L}(X)(u,\tau )=\operatorname{tr}(A^{u,\tau }X)\).
The range of the tail boundary map is exactly the tail ground space:
For the forward inclusion, if \(\psi =\Gamma ^{\mathrm{tail}}_{K,L}(Y)\) then \(R^{\mathrm{tail}}_u\psi =\Gamma _L(Y_u)\) for every prefix \(u\), so the suffix restriction lies in \(G_L(A)\). Conversely, given \(\psi \) with \(R^{\mathrm{tail}}_u\psi \in G_L(A)\) for every \(u\), choose for each \(u\) a matrix \(Y_u\in M_{D}(\mathbb {C})\) with \(\Gamma _L(Y_u)=R^{\mathrm{tail}}_u\psi \). Then \(\Gamma ^{\mathrm{tail}}_{K,L}(Y)(u,\tau )=\operatorname{tr}(A^\tau Y_u) =\Gamma _L(Y_u)(\tau )=(R^{\mathrm{tail}}_u\psi )(\tau )=\psi (u,\tau )\), so \(\psi \) lies in the range.
For every prefix \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) and suffix \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \),
Unfold the definition of \(\Gamma ^{\mathrm{left}}_{K,L}\) on the pair \((u,\tau )\), use \((\{ 0,\ldots ,.{-}1\} \! append\; u\; \tau )\circ \{ 0,\ldots ,.{-}1\} \! castAdd\; L=u\) and \((\{ 0,\ldots ,.{-}1\} \! append\; u\; \tau )\circ \{ 0,\ldots ,.{-}1\} \! natAdd\; K=\tau \), and apply \(\Gamma _K(Z_\tau )(u)=\operatorname{tr}(A^u Z_\tau )\).
Restricting the left boundary map to the prefix for a fixed suffix \(\tau \in \{ 0,\ldots ,d{-}1\} ^L\) recovers \(\Gamma _K(Z_\tau )\):
Pointwise application of Theorem 12.3.5.9: for each \(\sigma \), \((P_\tau \psi )(\sigma )=\psi (\sigma ,\tau )\).
The standard length-\((K+L)\) boundary map factors through the left boundary map:
for every \(X\in M_{D}(\mathbb {C})\).
Substitute \(Z_\tau =A^\tau X\) into the definition of \(\Gamma ^{\mathrm{left}}_{K,L}\), giving \(\operatorname{tr}(A^u(A^\tau X))=\operatorname{tr}((A^u A^\tau )X)\). Factor \(A^u A^\tau =A^{u,\tau }\) by Lemma 2.1.4 and identify the result with \(\Gamma _{K+L}(X)(u,\tau )\).
The range of the left boundary map is exactly the left ground space:
The proof is symmetric to Theorem 12.3.5.8. If \(\psi =\Gamma ^{\mathrm{left}}_{K,L}(Z)\) then \(P_\tau \psi =\Gamma _K(Z_\tau )\), so the prefix restriction lies in \(G_K(A)\). Conversely, given \(\psi \) with \(P_\tau \psi \in G_K(A)\) for every \(\tau \), choose \(Z_\tau \) with \(\Gamma _K(Z_\tau )=P_\tau \psi \); then \(\Gamma ^{\mathrm{left}}_{K,L}(Z)(u,\tau )=\operatorname{tr}(A^u Z_\tau ) =\Gamma _K(Z_\tau )(u)=(P_\tau \psi )(u)=\psi (u,\tau )\).
For \(L=1\), the prefix restriction \(P_\tau \) coincides with the last-site restriction: \(P_\tau \psi =R^{\mathrm{last}}(\psi ,\tau (0))\) for every \(\tau \colon \{ 0,\ldots ,1{-}1\} \to \{ 0,\ldots ,d{-}1\} \).
Unfolding both sides: for \(\sigma \in \{ 0,\ldots ,d{-}1\} ^K\), \((P_\tau \psi )(\sigma )=\psi (\sigma ,\tau )\) while \(R^{\mathrm{last}}(\psi ,j)(\sigma )=\psi (\sigma _1,\dots ,\sigma _K,j)\) with \(j=\tau (0)\). These are equal because \(\{ 0,\ldots ,.{-}1\} \! append\; \sigma \; \tau \) puts \(\tau \) after \(\sigma \), matching the \(\{ 0,\ldots ,.{-}1\} \! snoc\) pattern of \(R^{\mathrm{last}}\).
For every \(K,L\in \mathbb {N}\), transport of the tail boundary map range by the inverse canonical \(\ell ^2\) isomorphism identifies it with the open-chain tail ground submodule:
Concretely, a vector \(v\) in the \((K+L)\)-site Euclidean space belongs to \(\mathcal V_{K,L}(A)\) exactly when the corresponding function satisfies the tail ground condition (Definition 11.5.5.6).
By Theorem 12.3.5.8, the range equals the intersection over prefixes \(u\) of the ground-space comaps \((R^{\mathrm{tail}}_u)^{-1}(G_L(A))\). Transporting this intersection through the canonical \(\ell ^2\) identification and unfolding the definition of \(\mathcal V_{K,L}(A)\) (Definition 11.5.5.12) gives the stated equality.
At \(L=1\), transport of the left boundary map range by the inverse canonical \(\ell ^2\) isomorphism identifies it with the open-chain left ground submodule:
Concretely, a vector \(v\) in the \((K+1)\)-site Euclidean space belongs to \(\mathcal U_K(A)\) exactly when the corresponding function satisfies the left ground condition (every fixed-final-site restriction lies in \(G_K(A)\); Definition 11.5.1.4).
By Theorem 12.3.5.12 with \(L=1\), the range equals the intersection over suffixes \(\tau \colon \{ 0,\ldots ,1{-}1\} \to \{ 0,\ldots ,d{-}1\} \) of the comaps \(P_\tau ^{-1}(G_K(A))\). For \(L=1\), each suffix \(\tau \) is determined by a single index \(j\in \{ 0,\ldots ,d{-}1\} \), and Lemma 12.3.5.13 identifies \(P_\tau \) with \(R^{\mathrm{last}}(\cdot ,j)\), the last-site restriction of Definition 11.5.1.2. Transporting the resulting intersection \(\bigcap _{j\in \{ 0,\ldots ,d{-}1\} }(R^{\mathrm{last}}(\cdot ,j))^{-1}(G_K(A))\) through the canonical \(\ell ^2\) identification and unfolding Definition 11.5.5.12 gives \(\mathcal U_K(A)\).
Transport the tail and left boundary maps through the canonical Euclidean identifications on their virtual matrix families and physical coefficient spaces. Denote the resulting continuous maps by \(\Gamma ^{\mathrm{tail}}_{K,L}\) and \(\Gamma ^{\mathrm{left}}_{K,L}\).
Equip each virtual matrix fiber with the unweighted Hilbert–Schmidt inner product and flatten the finite family of fibers into a Euclidean space. The resulting continuous tail and left boundary maps have ranges \(\mathcal V_{K,L}(A)\) and, at suffix length one, \(\mathcal U_K(A)\). If \(J^{\mathrm{tail}}_K(X)_u=XA^u\) and \(J^{\mathrm{left}}_L(X)_\tau =A^\tau X\), then
Transport the algebraic range descriptions through the Euclidean identification. The two factorization identities follow by evaluating a virtual matrix \(X\) fiberwise and applying the corresponding algebraic tail or left factorization.
If the length-\(K\) ground-space map is injective, then the left boundary map with any spectator length \(L\) is injective.
Recover every spectator fiber by restricting to its fixed suffix and use injectivity of the length-\(K\) ground-space map.
The tail Gram is the direct sum over prefix configurations of copies of \(\mathcal K_L\), and the left Gram is the direct sum over suffix configurations of copies of \(\mathcal K_K\). Whenever the corresponding local boundary map is injective, the inverse Gram is the direct sum of the local inverse Grams. These identities include empty spectator index types and assert no norm estimate.
Reindex the physical inner product by prefix–suffix configurations and evaluate each fiber. This gives the direct-sum Gram identities. Composing fiberwise with the local inverse Gram on either side gives the stated inverse formulas.
Let \(S\) be a finite spectator index set, let \(V\) be the Euclidean virtual matrix space, and let \(G:V\to V\) be linear. If \(G^{\oplus S}\) acts on a boundary family \(x=(x_s)_{s\in S}\) by \((G^{\oplus S}x)_s=Gx_s\), then
This estimate also holds when \(S\) is empty.
Square both sides of the pointwise estimate \(\lVert Gx_s\rVert \leq \lVert G\rVert \lVert x_s\rVert \), sum over \(s\in S\), and use the Euclidean direct-sum norm identity.
Fix \(K,L_0\in \mathbb N\). Suppose that the tail boundary map for \((K,L_0+1)\), the left boundary map for \((K+L_0,1)\), and \(\Gamma _{K+L_0+1}\) are injective. The difference between the product of the first two inverse-Gram range projectors and the third inverse-Gram range projector is exactly the common-factorization residual obtained from the tail and left virtual maps. Their ranges are, respectively, \(\mathcal V_{K,L_0+1}(A)\), \(\mathcal U_{K+L_0}(A)\), and \(\mathcal G_{K+L_0+1}(A)\). This identity asserts no numerical contraction estimate.
Apply the common-factorization projector identity to the two spectator boundary maps, the full boundary map, and their tail and left virtual maps. These range equalities identify the three physical ranges.
Let \(q\in \mathbb N\), let \(X=(X_u)_{u\in \{ 0,\ldots ,d{-}1\} ^K}\), and let \(Z=(Z_j)_{j\in \{ 0,\ldots ,d{-}1\} ^q}\) be virtual boundary-matrix families. After the canonical identification of \(K+(l+q)\) with \((K+l)+q\), the mixed Gram for the tail window of length \(l+q\) and the left window with a \(q\)-site increment is
where \(U\) is Frobenius vectorization and \(\mathcal K_l=\Gamma _l^\dagger \Gamma _l\). The multiplication order is fixed by \(\operatorname{tr}(A^{\tau ,j}X_u)=\operatorname{tr}(A^\tau (A^jX_u))\) and \(\operatorname{tr}(A^{u,\tau }Z_j)=\operatorname{tr}(A^\tau (Z_jA^u))\). The one-site identity is the specialization \(q=1\). This identity expresses the algebraic form of condition (C3\('\)) in [ 225 , Equation (2.5) ] .
Split a physical configuration into a prefix \(u\), an overlap word \(\tau \) of length \(l\), and an increment configuration \(j\) of length \(q\). The adjoint relation turns the mixed Gram into the physical inner product of the two boundary vectors. For each \((u,\tau ,j)\), word multiplicativity and trace cyclicity give
For fixed \((u,j)\), summing the resulting inner products over \(\tau \) is exactly the boundary-map Gram pairing \(\langle U(A^jX_u),\mathcal K_l U(Z_jA^u)\rangle \). Summing over \(u\) and \(j\) proves the arbitrary-increment identity; setting \(q=1\) gives the original formula.
Specialize Theorem 12.3.5.22 to \(q=1\). Let \(X=(X_u)_{u\in \{ 0,\ldots ,d{-}1\} ^K}\) and \(Z=(Z_j)_{j\in \{ 0,\ldots ,d{-}1\} ^1}\) be the resulting boundary families. If \(\rho \in M_{D}(\mathbb {C})\) has nonzero trace and \(\mathcal K_\infty =\mathscr R(P_\rho )\), then
Here \(\mathcal K_\infty U(Z_jA^u) =U((\operatorname{tr}\rho )^{-1}Z_jA^u\rho )\), so the limiting multiplication is on the right and is generally not the identity.
Substitute the exact mixed-Gram formula and the right-multiplication action of \(\mathcal K_\infty \), then distribute subtraction through the two finite sums and the second argument of the inner product.
For arbitrary virtual dimension, the adjoints of the tail and left virtual maps are
If, in addition, \(D{\gt}0\) and \(\rho \in M_{D}(\mathbb {C})\) is positive definite, set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). Then
For the tail adjoint, test against an arbitrary matrix \(Y\) and use the Hilbert–Schmidt trace pairing fiberwise:
The left formula follows in the same way from \(\langle U(A^\tau Y),U(Z_\tau )\rangle =\langle U(Y),U((A^\tau )^\dagger Z_\tau )\rangle \).
Under the additional positive-definiteness hypothesis, the forward action of \(\mathcal K_\infty \) is the exact limiting-metric formula and the operator is invertible. Applying the forward formula to \(U((\operatorname{tr}\rho )X\rho ^{-1})\) cancels both scalar factors and the two right matrix factors, which gives the displayed inverse action.
Let \(\rho \in M_{D}(\mathbb {C})\) be positive definite and set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). Write \(J_{\rm t}\) and \(J_{\rm l}\) for the tail and left virtual maps, and write \(S_n=\mathcal K_n^{-1}\) whenever the length-\(n\) boundary map is injective. The virtual residual in the exact tail-after-left C3 projector factorization has the telescoping decomposition
where \(M_l=(\Gamma ^{\rm tail}_{K,l+1})^\dagger \Gamma ^{\rm left}_{K+l,1}\) and the superscript \(\oplus \) denotes the corresponding fiberwise spectator operator. Substituting this equality into the exact inverse-Gram factorization gives the same decomposition of the physical projector residual, sandwiched between the tail boundary map, the two local fiberwise inverse Grams, and the adjoint left boundary map.
The limiting product in the first parenthesis has, without an additional hypothesis, the explicit limiting mixed pairing
Consequently, pairing the first parenthesis against boundary families is exactly the same finite sum with \(\mathcal K_l-\mathcal K_\infty \) in place of \(\mathcal K_l\).
These identities are reconstructed operator algebra. They assert neither the numerical FNW/Nachtergaele contraction nor any identification of a physical ground projector with \(P_\rho \).
Add and subtract the product in which all three finite Gram factors are replaced by their limiting values, and telescope the resulting product from left to right. Substitute the direct-sum Gram and inverse-Gram formulas into the exact projector factorization.
It remains to identify the leading limiting product. In Frobenius coordinates the limiting metric and its inverse act by
The adjoint left virtual map sends a family \(W=(W_j)_j\) to \(U(\sum _j(A^j)^\dagger W_j)\). For \(W_j=(\operatorname{tr}\rho )^{-1}Z_j\rho \), the right multiplication by \(\rho \) and the factor \((\operatorname{tr}\rho )^{-1}\) supplied by the left limiting spectator metric cancel exactly through \(S_\infty \). Applying the tail virtual map and the tail limiting spectator metric leaves, in the fiber indexed by \(u\),
Its inner product with \(X_u\), followed by summation over \(u\), is the displayed limiting mixed pairing by the Frobenius trace identity. No transfer fixed-point equation is needed for this zeroth-order cancellation. The centered mixed-Gram identity then turns the leading term into the pairing with \(\mathcal K_l-\mathcal K_\infty \).
12.3.6 Martingale gap consequences
Let \(x\) and \(y\) belong to a complex inner-product space. For every \(c{\gt}0\),
Cauchy–Schwarz bounds the left-hand side by \(\lVert x\rVert \lVert y\rVert \). Apply the weighted arithmetic–geometric mean inequality to \(\lVert y\rVert \) and \(\lVert x\rVert \).
Let \(P\) be an orthogonal projection on a complex inner-product space, and let \(Q\) be a bounded linear operator from that space to another. If
then, for every \(v\),
This includes \(\epsilon =0\).
Idempotency gives \(QPv=QP(Pv)\). Apply the operator-norm estimate for \(QP\) to \(Pv\) and square the resulting nonnegative inequality. The hypothesis itself implies \(\epsilon \geq 0\).
Let \((a_m)_{m\geq 0}\) be a nonnegative real sequence. For all \(l,N\in \mathbb {N}\),
The same bound holds after omitting the terms with \(n{\lt}l\). More generally, for a threshold \(n_0\in \mathbb {N}\) only the coefficients indexed by \(m\geq \max \{ 0,n_0-l\} \) occur, so
Write \(k=n-m\). Each pair in the double sum has \(0\leq k\leq l\). For fixed \(k\), the map \(n\mapsto n-k\) is injective and has image in \(\{ 0,\ldots ,N-1\} \). Nonnegativity bounds its contribution by \(\sum _{m=0}^{N-1}a_m\). Summing over the \(l+1\) values of \(k\) proves (128). This is the counting step in the proof of Nachtergaele’s Theorem 2.1(i), between Equation (Enpsi2) and the following summed estimate: for \(a_m=\lVert E_m\psi \rVert ^2\), each term occurs in at most \(l+1\) consecutive windows [ 225 , Proof of Theorem 2.1(i) ] . For (129), replace \(a_m\) by the sequence that vanishes below \(\max \{ 0,n_0-l\} \) and agrees with \(a_m\) above it. Every window with \(n\geq n_0\) is unchanged, and the previous bound applied to the replacement sequence gives the claim.
Let \((G_n)_{n\geq 0}\) be nested orthogonal projections on a finite-dimensional complex Hilbert space \(\mathcal H\), put \(E_n=G_n-G_{n+1}\), and assume \(G_0=\mathbf1\). Fix \(l,N,n_0\in \mathbb {N}\) with \(n_0\leq N\), and let \(Q_n\) be orthogonal projections for \(n_0\leq n{\lt}N\). Suppose \(E_mQ_n=Q_nE_m\) whenever \(m{\lt}n-l\) or \(n{\lt}m\). Let \(H_n\) be the local Hamiltonians and \(H\) the full Hamiltonian. Assume the quadratic-form version of C1 on the same range, for every \(x\in \mathcal H\),
the quadratic-form version of C2 for every \(x\in \mathcal H\) and \(n_0\leq n{\lt}N\),
and the literal condition C3 for \(n_0\leq n{\lt}N\),
If \(\gamma _{l+1}{\gt}0\), \(d_{l+1}{\gt}0\), and \(v\perp \operatorname {ran}(G_N)\), then
Conditions C2 and C3 have the source forms, with their onset represented by \(n_0\). The displayed C1 inequality follows from source C1 when \(l\leq n_0\) and the omitted local terms are nonnegative. Thus a source-derived application chooses \(n_0\) above both \(l\) and the C2–C3 onset. If \(n_0{\lt}l\), the displayed C1 inequality is instead an independent stronger assumption. The coefficient of the martingale mass above the threshold is exactly the one in Theorem 2.1(i) of [ 225 ] . The correction is an explicit upper bound for the contribution of the at most \(l\) martingale differences immediately below the threshold; it uses multiplicity \(l+1\) uniformly. At \(n_0=0\) the correction is an empty sum.
Write \(P=\sum _{n=n_0}^{N-1}\lVert E_nv\rVert ^2\) and \(S=\sum _{m=\max \{ 0,n_0-l\} }^{n_0-1}\lVert E_mv\rVert ^2\). The martingale resolution \(v=\sum _{n=0}^{N-1}E_nv\) and outside-window cancellation reduce the identity labelled Enpsi at index \(n\) to the active window \(m\in [n-l,n]\); this uses only \(n{\lt}N\), so it is available for every \(n\geq n_0\). Apply the weighted inner-product estimate with positive parameters \(c_1,c_2\), then C2 and C3, and sum over \(n_0\leq n{\lt}N\). The martingale differences are mutually orthogonal, so each window norm square splits into its summands, and (129) bounds the resulting double sum by \((l+1)(S+P)\). With C1 this gives
For \(\epsilon _l{\gt}0\), take \(c_1=1-\epsilon _l\sqrt{l+1}\) and \(c_2=\epsilon _l/\sqrt{l+1}\), so that \(\epsilon _l^2/c_2=(l+1)c_2=\epsilon _l\sqrt{l+1}\) and the left factor is \(1-\epsilon _l\sqrt{l+1}\). Multiplying by \(c_1\gamma _{l+1}/d_{l+1}\) gives (133). If \(\epsilon _l=0\), then \(c_2=0\) is not an admissible weighted parameter. In that case C3 instead gives \(Q_nE_n=0\), so the second inner product in Enpsi vanishes; applying only the first weighted estimate with \(c_1=1\) yields \(P\leq (d_{l+1}/\gamma _{l+1})\operatorname{Re}\langle Hv,v\rangle \), which is the same formula because the correction term vanishes. This repair is documented in [ 135 ] .
Let \((G_n)_{n\geq 0}\) be nested orthogonal projections on a complex Hilbert space, and put \(E_n=G_n-G_{n+1}\). If \(G_{n_0}v=v\) for some \(n_0\), then \(E_nv=0\) for every \(n{\lt}n_0\).
For \(m\leq n_0\) the ranges are nested, so \(G_mG_{n_0}=G_{n_0}\) and hence \(G_mv=G_mG_{n_0}v=G_{n_0}v=v\). For \(n{\lt}n_0\) this applies to both \(n\) and \(n+1\), so \(E_nv=G_nv-G_{n+1}v=v-v=0\).
Assume the hypotheses of Theorem 12.3.6.4, except that \(l\leq n_0\) is assumed and (130) is replaced by condition C1 on the range \(l\leq n{\lt}M\) of its own window sum at the two volumes at which the source asserts it: the full inequality \(0\leq \sum _{n=l}^{N-1}\operatorname{Re}\langle H_nx,x\rangle \leq d_{l+1}\operatorname{Re}\langle Hx,x\rangle \) at \(M=N\), and its non-negativity clause \(0\leq \sum _{n=l}^{n_0-1}\operatorname{Re}\langle H_nx,x\rangle \) at \(M=n_0\). Assume in addition that \(E_nv=0\) for every \(n{\lt}n_0\). Then
This is the conclusion of Theorem 2.1(i) of [ 225 ] with each of C1, C2, and C3 imposed on the range at which the source imposes it. The added vanishing hypothesis is the minimal repair of that theorem: the two below-threshold quantities in (133) enter (135) with non-negative weights, so both have to vanish, and the vanishing of the omitted martingale mass forces the vanishing of the window correction. Without the added hypothesis the statement is false, by Theorem 12.3.6.7. The source defect and the repair are recorded in [ 134 ] .
Condition C1 on the threshold range \(n_0\leq n{\lt}N\) follows from the assumed form: its upper bound by subtracting from the sum over \(l\leq n{\lt}N\) the initial sum over \(l\leq n{\lt}n_0\), which is non-negative by C1 at the volume \(\Lambda _{n_0}\), and its non-negativity clause termwise from C2, each of whose left-hand sides is non-negative. The window correction in (133) is a sum of squared norms of martingale differences with index below \(n_0\), hence zero. The Pythagorean decomposition of a vector orthogonal to \(\operatorname {ran}(G_N)\) splits \(\lVert v\rVert ^2\) into the sum over \(n{\lt}n_0\), which is zero for the same reason, and the martingale mass \(\sum _{n=n_0}^{N-1}\lVert E_nv\rVert ^2\). Hence the latter equals \(\lVert v\rVert ^2\), and (133) becomes (135).
In the notation of Theorem 12.3.6.6: for every \(k\) there are, on a complex Hilbert space of dimension \(k+2\), nested orthogonal projections \((G_n)_{n\geq 0}\) with \(G_0=\mathbf1\), \(G_{N+1}=0\), and \(G_n\neq 0\) for every \(n\leq N\), nonzero orthogonal projections \(Q_n\), local operators \(H_n\), volume Hamiltonians \(H_{\Lambda _M}\), and a vector \(v\perp \operatorname {ran}(G_N)\) such that, with \(l=0\), \(n_0=1\), \(N=k+1\), \(\gamma _1=d_1=1\), and \(\epsilon _0=0\), the required commutations hold, condition C1 holds with the single constant \(d_1\) at every volume \(\Lambda _M\) on the range \(l\leq n{\lt}M\) carried by its own window sum, the gap inequality of condition C2 and condition C3 hold on the range \(n_0\leq n{\lt}N\) from which they are imposed, every ground space is the kernel of its own Hamiltonian, that is \(\ker H_{\Lambda _M}=\operatorname {ran}(G_M)\) for \(M\leq N\) and \(\ker H_n=\operatorname {ran}(Q_n)\) for every window inside \(\Lambda _N\), that is for \(0\leq n{\lt}N\), where the \(Q_n\) are also nonzero orthogonal projections commuting with the martingale differences outside their own windows, and
Hence condition C1 at every volume on the range of its own window sum together with conditions C2 and C3 on \(n_0\leq n{\lt}N\), the ranges at which [ 225 ] imposes them, do not imply the conclusion of its Theorem 2.1(i), even when every ground space is the kernel of its Hamiltonian and is nontrivial, as condition C2 also requires, and even under the conventions the source attaches to the extreme volumes.
Fix an orthonormal basis \(e_0,\ldots ,e_{k+1}\) and let every operator be diagonal in it. The first \(N=k+1\) basis vectors carry the single excitations of a chain of \(N\) sites and the last carries its ground state. Let \(G_n\) have weight \(1\) at \(e_i\) when \(n\leq i\) and weight \(0\) otherwise, so that \(G_0=\mathbf1\), \(G_{N+1}=0\) because no basis vector has index at least \(N+1\), every \(G_n\) with \(n\leq N\) is nonzero because it retains \(e_{k+1}\), \(G_N\) is the projection onto \(e_{k+1}\), and \(E_n\) is the projection onto \(e_n\) for \(n{\lt}N\). Let \(c_0=\tfrac 12\), let \(c_i=1\) for \(1\leq i\leq k\), and let \(c_{k+1}=0\); let \(H_{\Lambda _M}\) have weight \(c_i\) at \(e_i\) when \(i{\lt}M\) and weight \(0\) otherwise, let \(H_n\) have weight \(c_n\) at \(e_n\) and weight \(0\) elsewhere, and let \(Q_n=\mathbf1-E_n\), which is nonzero because the space has dimension at least two. All operators are diagonal, so the required commutations hold, and \(Q_nE_n=0\) gives condition C3 with \(\epsilon _0=0\). Summing \(H_n\) over the C1 range \(0\leq n{\lt}M\) gives \(H_{\Lambda _M}\) exactly, at every \(M\), and the weights \(c_i\) are nonnegative, so condition C1 holds at every volume with the single constant \(d_1=1\) and with equality in its upper bound. Since \(c_i\neq 0\) for \(i\leq k\), the kernel of \(H_{\Lambda _M}\) is spanned by the \(e_i\) with \(i\geq M\) whenever \(M\leq N\), which is \(\operatorname {ran}(G_M)\), and the kernel of \(H_n\) is spanned by the \(e_i\) with \(i\neq n\) whenever \(0\leq n{\lt}N\), which is \(\operatorname {ran}(Q_n)\) and is nonzero. For \(1\leq n{\lt}N\) one has \(c_n=1\), so \(\lVert (\mathbf1-Q_n)x\rVert ^2=\lVert E_nx\rVert ^2\) equals \(\operatorname{Re}\langle H_nx,x\rangle \), which is condition C2 with \(\gamma _1=1\). Take \(v=e_0\). Then \(v\perp \operatorname {ran}(G_N)\), the line through \(e_{k+1}\), and \(\operatorname{Re}\langle H_{\Lambda _N}v,v\rangle =c_0=\tfrac 12\), while \(\lVert v\rVert ^2=1\), which is (136). The whole martingale mass of \(v\) sits at the index \(n=0\), which C1 covers while C2 and C3 exclude it, and it is at that index that C2 fails.
Let \((G_n)_{n\geq 0}\) be nested orthogonal projections on a finite-dimensional complex Hilbert space \(\mathcal H\), put \(E_n=G_n-G_{n+1}\), and assume \(G_0=\mathbf1\). Fix \(l,N\in \mathbb {N}\), and let \(Q_n\) be orthogonal projections for \(0\leq n{\lt}N\). Suppose \(E_mQ_n=Q_nE_m\) whenever \(m{\lt}n-l\) or \(n{\lt}m\). Let \(H_n\) be the local Hamiltonians and \(H\) the full Hamiltonian. Assume the quadratic-form version of C1 for every \(x\in \mathcal H\),
and the quadratic-form version of C2 for every \(x\in \mathcal H\) and \(0\leq n{\lt}N\),
and the literal condition C3,
If \(\gamma _{l+1}{\gt}0\), \(d_{l+1}{\gt}0\), and \(v\perp \operatorname {ran}(G_N)\), then
This full-range statement is stronger than the unrestricted source theorem in general. It assumes C1, C2, and C3 for every \(0\leq n{\lt}N\), whereas the source imposes lower thresholds on these conditions. Those thresholds cannot simply be dropped: without separate control of the initial volumes the source hypotheses imply neither the assumptions above nor the conclusion, by Theorem 12.3.6.7. The coefficient is exactly the one in Theorem 2.1(i) of [ 225 ] : \(\gamma _{l+1}/d_{l+1}\) times \((1-\epsilon _l\sqrt{l+1})^2\). This is Theorem 12.3.6.4 at threshold \(n_0=0\), where nothing lies below the threshold.
Apply Theorem 12.3.6.4 with \(n_0=0\) and the given window length \(l\). Its window correction is a sum over an empty range, hence zero, and its martingale mass over \(0\leq n{\lt}N\) is \(\lVert v\rVert ^2\) by the Pythagorean decomposition of a vector orthogonal to \(\operatorname {ran}(G_N)\). What remains is (140).
Let \(H\) be a linear operator on a complex Hilbert space. If
for every \(v\in (\ker H)^\perp \), then \(\kappa \lVert v\rVert \leq \lVert Hv\rVert \) on the same subspace.
Cauchy–Schwarz gives
The claim is immediate for \(v=0\); otherwise cancel \(\lVert v\rVert \).
In the setting of Theorem 12.3.6.8, suppose that \(H\) is positive and \(\operatorname {ran}(G_N)=\ker H\). Then every \(v\in (\ker H)^\perp \) satisfies
Denote the coefficient in (143) by \(\kappa \). For an arbitrary \(x\), remove its orthogonal projection onto \(\ker H\) and call the remainder \(w\). Then \(Hx=Hw\) and \(\operatorname{Re}\langle Hx,x\rangle =\operatorname{Re}\langle Hw,w\rangle \). The energy estimate and Theorem 12.3.6.9 give \(\kappa \lVert w\rVert \leq \lVert Hx\rVert \). Hence Cauchy–Schwarz gives
Thus the hypothesis of Lemma 12.3.20 holds globally. The finite-dimensional quadratic-form criterion then yields the stated norm bound.
Let \(P_i\) be a finite family of symmetric projections on a finite-dimensional complex Hilbert space, let \(H=\sum _iP_i\), and let \(0{\lt}\gamma \le 1\). Suppose that a real coefficient matrix \((c_{ij})\) has row and column sums at most one away from the diagonal, and that, for \(i\ne j\) and every \(v\),
Then
for every \(v\). Consequently, for every \(v\in (\ker H)^\perp \),
The row-and-column reduction gives the quadratic-form inequality. The sum of symmetric projections is positive, so the abstract martingale criterion converts that inequality into the norm lower bound on \((\ker H)^\perp \).
Let \(U\) and \(V\) be ground-space ranges in a finite-dimensional complex Hilbert space, let \(W=U\cap V\), and let \(q_U\) and \(q_V\) be the orthogonal projectors onto \(U\) and \(V\). Suppose that \(\eta \ge 0\) and
whenever \(x\in U\cap W^\perp \) and \(y\in V\cap W^\perp \). Define the complementary excitation projections by \(h_U=\mathbb {1}-q_U\) and \(h_V=\mathbb {1}-q_V\). Then, for every vector \(v\),
Remove the component of \(v\) in the common intersection \(W\); both excitation projections are unchanged. If \(\eta \le 1\), apply (147) to the reduced ground-space components \(q_Uv\) and \(q_Vv\), and combine the resulting principal-angle estimate with the orthogonal decompositions \(v=q_Uv+h_Uv=q_Vv+h_Vv\). If \(\eta {\gt}1\), the same conclusion follows from the universal overlap bound with coefficient \(1\) and the non-negativity of the two diagonal excitation forms.
12.3.7 The compatible open-chain Hamiltonian
For an interval, the open-chain parent Hamiltonian contains only local terms whose supports lie entirely within the interval [ 225 , Equation (3.12) ] . The corresponding zero-energy space coincides with the open-chain ground space. Each compatible open Hamiltonian is positive as a sum of orthogonal projections. Conditions (C1) and (C2) hold across the full martingale range: before a complete interaction window fits, the suffix Hamiltonian vanishes, and on larger intervals it equals the orthogonal projection onto the local excited space. Combined with condition (C3), these identities specialize Theorem 12.3.6.10 with \(d_{l+1}=\gamma _{l+1}=1\) and overlap bound \(\epsilon _l\).
For \(L,N\in \mathbb {N}\), let
The compatible length-\(N\) open-chain parent Hamiltonian of interaction length \(L\) is
Thus no summand crosses the two endpoints of the interval. For \(L=l+1\), the starting sites are \(0,\ldots ,N-l-1\), in agreement with [ 225 , Equation (3.12) ] .
If \(L{\gt}0\), then
The interaction has one window:
The outside region is empty, so it has a unique configuration \(\tau \). Cyclic restriction to the sole window is the identity:
Likewise,
Therefore
If \(L{\gt}0\) and \(v\in (\ker H^{\mathrm{open}}_{L,L}(A))^\perp \), then
In particular,
Thus \(\gamma _{l+1}=1\) at the full-window volume \(N=L=l+1\) in [ 225 , Theorem 2.1(i) ] . This neither asserts that the excited subspace is nonzero nor supplies a gap for any \(N{\gt}L\).
The full-window identity and the local kernel formula give
Hence \(v\) belongs to \((G_L^{\mathrm{ES}}(A))^\perp \), where the orthogonal projector \(P_L^{\mathrm{ES}}(A)\) acts as the identity. Thus
Consequently,
For every \(A\), \(L\), and \(N\), the operator \(H^{\mathrm{open}}_{N,L}(A)\) is positive.
Each \(h_{i,\mathrm{ES}}(A,L)\) is an orthogonal projection and hence is positive. Positivity is preserved by finite sums.
For \(L,l,n,N\in \mathbb {N}\), the Hamiltonian on the last \(l\) sites of the prefix \([0,n)\), regarded as an operator on the \(N\)-site chain, is
Its local windows lie in the nonwrapping interval \([n-l,n)\).
If \(n{\lt}L\), then
A summand would require a starting site \(i\) satisfying both \(n-L\leq i\) and \(i+L\leq n\). The latter inequality is impossible when \(n{\lt}L\) and \(i\geq 0\), so the defining sum is empty.
If \(L{\gt}0\), then
The substitution \(m=n+1\) identifies the left-hand side with the sum over \(1\leq m\leq N\). Every term with \(m{\lt}L\) vanishes by Theorem 12.3.7.6, leaving precisely the stated interval.
If \(L\leq l\), then
Thus condition C1 of [ 225 ] holds with \(d_l=l-L+1\). This is the source value \(pl-r+1\) in the present convention \(p=1\) and interaction range \(r=L\).
Fix a starting site \(i\in I_{L,N}\). The suffix Hamiltonian indexed by \(n\) contains its local term exactly when
This interval contains at most \(l-L+1\) integers. Reversing the two finite sums therefore writes the left-hand side as
where \(0\leq c_i\leq l-L+1\). Positivity of each local term turns the coefficient bound into the upper operator inequality. Positivity of their sum gives the lower inequality.
Set \(L=l+1\). For every vector \(v\) on the \(N\)-site chain,
Thus the full martingale range satisfies C1 with \(d_{l+1}=1\).
Let \(L{\gt}0\) and \(L\leq m\leq N\). The suffix of length \(L\) ending at \(m\) contains one interaction:
Write \(G^{\ker }_{N,L;m}\) for the orthogonal projection onto the kernel of this suffix Hamiltonian. Then
Consequently condition C2 holds in Loewner order with \(\gamma _L=1\):
Equivalently, for every \(v\),
If \(v\in (\ker H^{\mathrm{suffix}}_{N,L;L,m}(A))^\perp \), then
In particular,
In the notation of [ 225 ] , set \(L=l+1\) and \(m=n+1\). The interval is then \(\Lambda _{n+1}\setminus \Lambda _{n-l}=[n-l,n+1)\), so
Condition C2 has the constant
The later condition C3 uses the distinct overlap parameter \(\epsilon _l\).
An admissible start \(i\) satisfies
It also satisfies
Hence \(i=m-L\), which proves (169). The sole local term is an orthogonal projection. For any orthogonal projection \(H\),
Therefore
This gives (170). The Loewner and quadratic-form inequalities are equalities. On \((\ker H)^\perp =\operatorname {ran}H\), idempotence gives \(Hv=v\), and hence exact norm preservation and the unit lower bound. Condition C1 at suffix length \(L\) gives \(d_L=L-L+1=1\). With \(L=l+1\), the two constants are therefore \(d_{l+1}=1\) and \(\gamma _{l+1}=1\).
Set \(L=l+1\), let \(0\leq n{\lt}N\), and let \(Q_n\) be the orthogonal projection onto the kernel of \(H^{\mathrm{suffix}}_{N,L;L,n+1}(A)\). Then every vector \(v\) satisfies
Hence C2 holds on the full martingale range with \(\gamma _{l+1}=1\).
If \(n{\lt}l\), the suffix Hamiltonian is zero and \(Q_n=I\), so both sides of (181) vanish. If \(n\geq l\), the suffix contains the single length-\((l+1)\) interaction and equals \(I-Q_n\) by Theorem 12.3.7.10. Since \(I-Q_n\) is an orthogonal projection,
If \(H^{\mathrm{open}}_{N,L}(A)v=0\), then, for every \(i\in I_{L,N}\),
Taking the inner product of the sum with \(v\) gives
Each summand is the squared norm of the image of \(v\) under an orthogonal projection. Hence every local term annihilates \(v\).
Assume \(s+L\leq N\). For every boundary matrix \(X\) and complementary configuration \(\tau \),
For the fixed words
and
the word of a configuration obtained by inserting the window word \(\omega \) is
Cyclicity of the trace gives
and hence
For every \(A\), \(L\), and \(N\),
Let \(v=\iota _N(\Gamma _N^A(X))\in G_N^{\mathrm{ES}}(A)\). For every non-wrapping window \([s,s+L-1]\) and complementary configuration \(\tau \), Theorem 12.3.7.13 gives
The non-wrapping identification and Lemma 11.3.2 then give
The local kernel criterion therefore gives
Thus every compatible local term annihilates \(v\).
Let \(L\le N\). If \(H^{\mathrm{open}}_{N,L}(A)v=0\), then for every nonwrapping start \(s\) with \(s+L\le N\) and every complementary configuration \(\tau \),
This is the frustration-free extraction in [ 225 , Equation (3.12) ] .
A kernel vector of a sum of orthogonal projections is a kernel vector of each summand. The local kernel criterion then places every window restriction in the local ground space.
Assume \(D\geq 1\), \(L_0{\gt}0\), and \(N\geq L_0+1\). If \(A\) is \(L_0\)-block-injective, then
In the following display, \(s\) ranges over all starts satisfying \(s+L_0+1\leq N\), and \(\tau \) ranges over all complementary configurations. The window restriction and the open-chain intersection theorem give
Assume \(D\geq 1\), \(L_0{\gt}0\), and \(N\geq L_0+1\). If \(A\) is \(L_0\)-block-injective, then
This is the canonical MPS instance of the ground-state property in [ 225 , Equation (3.13) ] . The right-hand side is the open boundary-condition space and may therefore be degenerate.
For every tensor \(A\) and every \(L\), the local ground space \(G_{L}(A)\) is spanned by the open-boundary states
Here \((l|\) and \(|r)\) are the boundary vectors on both sides of [ 17 , line 1174 ] , written with the bilinear pairing.
The rank-one boundary matrix \(|r)(l|\) gives \(\Gamma _{L}(|r)(l|)(\sigma )=\operatorname{tr}(A^{\sigma }|r)(l|) =(l|A^{\sigma }|r)\). Every boundary matrix is a linear combination of matrix units, and each matrix unit is rank one.
If \(A\) is \(L\)-block-injective, then \(\dim G_{L}^{\mathrm{ES}}(A)=D^{2}\). If \(A\) is \(L_{0}\)-block-injective with \(L_{0}{\gt}0\), \(D\geq 1\) and \(N\geq L_{0}+1\), then
The local support spaces of a pure generalized valence-bond-solid state with auxiliary algebra \(M_{k}(\mathbb {C})\) have dimension \(k^{2}\) on long intervals, and they are the kernels of the open-interval Hamiltonian [ 225 , lines 1505–1510 and 1545–1558 ] ; the growth argument on an open chain is [ 228 , lines 1211–1233 ] . The review states that the open-boundary degeneracy is at least the square of the dimension of the irreducible projective representation carried by the boundary [ 17 , line 1174 ] . That dimension is at most \(D\), so the exact count \(D^{2}\) is sharper than the review’s lower bound.
Assume \(D\geq 1\), \(l{\gt}0\), and that \(A\) is \(l\)-block-injective. Fix \(K{\gt}0\), set \(n=K+l\) and \(n_l=l+1\), and work in the Hilbert space on \(\Lambda _{n+1}\). The physical open-chain martingale difference is
The first projector imposes the ground condition on the left \(n\) sites while acting on all \(n+1\) sites. Block injectivity identifies these MPS spaces with the corresponding open-parent Hamiltonian kernels. The condition \(K{\gt}0\) excludes the undersized volume, where an interaction of length \(l+1\) has no nonwrapping window and its physical ground projector is the identity.
For every \(K,l\in \mathbb {N}\), set \(n=K+l\). Then
Every fixed-prefix suffix restriction of a whole-chain ground-space vector is a ground-space vector on the suffix. This is exactly membership in \(\mathcal V_{K,l+1}(A)\).
For every \(K,l\in \mathbb {N}\), the tail projector fixes the whole-ground projector:
The inner projection has range in \(\mathcal G_{n+1}^{\mathrm{ES}}(A)\), which lies in the tail ground space by Lemma 12.3.7.21. Orthogonal projection onto that tail space therefore leaves every vector in the inner range unchanged.
Assume \(D\geq 1\), \(l{\gt}0\), and that \(A\) is \(l\)-block-injective. Fix \(K{\gt}0\). With \(n=K+l\) and \(n_l=l+1\), the MPS parent chain satisfies Nachtergaele’s identity following condition C3:
Distribute the tail projector over \(E_n=G_{\Lambda _n}-G_{\Lambda _{n+1}}\). The second product is \(G_{\Lambda _{n+1}}\) by Lemma 12.3.7.22, which gives the displayed identity.
Assume \(D\ge 1\) and let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Set \(n=K+L_0\) and \(l=L_0\), and write
Thus \(S\) and \(T\) are the left and tail ground spaces in the Hilbert space on \(\Lambda _{n+1}\). Let \(q_S=P_{S^\perp }\) and \(q_T=P_{T^\perp }\) be the complementary excitation projections. If
then every vector \(v\) in the Hilbert space on \(\Lambda _{n+1}\) satisfies
The open-chain intersection theorem identifies \(S\cap T\) with the ground space on \(\Lambda _{n+1}\). The projector-defect adapter gives a Friedrichs overlap bound, and the reduced two-projection theorem gives the stated anticommutator inequality.
Let \(A\) be a primitive MPS tensor with nonzero virtual dimension, and let \(J_K=J^{\mathrm{tail}}_K\) be its length-\(K\) tail virtual map. Then there is a constant \(C{\gt}0\) such that, for every \(K\in \mathbb {N}\),
No positive-definiteness hypothesis on the fixed point is required.
The operator norm is controlled using the adjoint identity
The Hilbert–Schmidt product estimate then gives the fourth-power bound
For \(K\geq 1\), decompose the transfer power into its fixed-point projection and the \(K\)-th power of the complementary map. The complementary powers are geometrically bounded by Theorem 11.4.4, while the entrywise mass is controlled by the \(L^2\) operator norm with the virtual dimension factor from Theorem 11.3.27. The case \(K=0\) is bounded directly. This first gives a uniform bound for \(\lVert J_K\rVert ^4\) and hence a strictly positive uniform bound for \(\lVert J_K\rVert \). No FNW or Nachtergaele contraction estimate enters.
If \(A\) is left-canonical, then for every \(Q\in \mathbb {N}\),
where \(\Omega _Q=\{ 0,\ldots ,({-}1\} d^Q)\) is the set of physical words of length \(Q\).
Induct on \(Q\). Split a word of length \(Q+1\) into its first letter and its remaining suffix, apply the induction hypothesis to the suffix sum, and then use left-canonicity for the outer one-site sum.
If \(A\) is left-canonical, then the length-\(Q\) left virtual map \(J_Q^{\mathrm{left}}\) preserves the Hilbert–Schmidt norm. In particular,
Expand the squared norm as a sum over \(j\in \Omega _Q\). The identity from Theorem 12.3.7.26 reduces the sum to the Hilbert–Schmidt norm of \(X\). The operator-norm bound follows immediately.
For \(K,L,Q\in \mathbb {N}\), let
be the unitary coordinate map induced by associativity of addition.
For every \(K,L,Q\in \mathbb {N}\),
Evaluate both sides at a physical word. Reassociation changes only the parenthesization of its coordinate set, so both coefficients are the trace pairing with the same word product.
Define the tail boundary map in the common \((K+L+Q)\)-site coordinates by
A one-site suffix increment needs no reassociation:
Reassociating a window whose last block carries a single site is the identity, so the reassociated tail map is the ordinary tail map with \(L+1\) spectator sites.
Let \(K,L,Q\in \mathbb {N}\) and let \(Y\colon (\{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} )\to M_{D}(\mathbb {C})\) be a boundary family. For all \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \), \(w\colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \), and \(v\colon \{ 0,\ldots ,Q{-}1\} \to \{ 0,\ldots ,d{-}1\} \),
The reassociation map only regroups the physical coordinates: for every vector \(f\) on \(K+(L+Q)\) sites,
Applying this to \(f=\Gamma ^{\mathrm{tail}}_{K,L+Q}(Y)\) and then Theorem 12.3.5.5 with suffix \((w,v)\) of length \(L+Q\) gives
The reassociated tail map satisfies
Substitute the definition of the reassociated tail map, use the ordinary tail factorization, and then reassociate the full boundary map.
Let \(T=\widetilde\Gamma ^{\mathrm{tail}}_{K;L,Q}\), \(S=\Gamma ^{\mathrm{left}}_{K+L,Q}\), and \(C=\Gamma _{K+L+Q}^{\mathrm{ES}}\). Assume separately that \(T\), \(S\), and \(C\) are injective. Since \(TJ_K^{\mathrm{tail}}=C=SJ_Q^{\mathrm{left}}\), the difference \(P_TP_S-P_C\) equals the exact inverse-Gram residual obtained from these two factorizations.
Apply the common-factorization range-projector identity to \(T\), \(S\), and \(C\), using the two displayed factorizations. No canonicality or norm estimate is used.
With the notation of Theorem 12.3.5.22,
Expand the reassociated tail map. Its adjoint introduces \(\mathscr A_{K,L,Q}^{-1}\), so the coordinate formula of Theorem 12.3.5.22 applies.
Let \(\rho \in M_{D}(\mathbb {C})\) have nonzero trace and set \(\mathcal K_\infty =\mathscr R(P_\rho )\). Then
Substitute the reassociated mixed-Gram formula and the action of the limiting metric, then distribute subtraction over the finite sums.
For \(X=(X_u)_{u\in \Omega _K}\), define
For \(Z=(Z_j)_{j\in \Omega _Q}\), define
The matrix fiber of \(F_{K,Q}X\) at \((u,j)\) is \(A^jX_u\).
Unfold the direct-sum factor and the Frobenius coordinate identification.
The matrix fiber of \(G_{K,Q}Z\) at \((u,j)\) is \(Z_jA^u\).
Unfold the direct-sum factor and the Frobenius coordinate identification.
The overlap factors satisfy
Expand the squared direct-sum norms fiberwise and apply the operator-norm bound to each fiber. Summing gives the two inequalities without a factor depending on the number of spectator words.
Assume \(\operatorname {tr}(\rho )\neq 0\) and let \(\mathcal K_\infty =\mathscr R(P_\rho )\). Define
The centered residual factors as
Test both sides against arbitrary boundary families, use the two fiber formulas, and apply the centered mixed-Gram identity.
The factorization gives
Apply submultiplicativity to the three factors and use the operator norm of a fiberwise direct sum.
If \(A\) is left-canonical, then
Let \(A\) be a primitive MPS tensor with positive-definite fixed point \(\rho \). Then \(A\) satisfies the left-canonical centered residual estimate of Theorem 12.3.7.45.
Use the left-canonical normalization contained in the primitive-MPS hypotheses together with positive definiteness of \(\rho \).
Let \(A\) be an MPS tensor, let \(\Omega _K\) be the set of length-\(K\) words in the physical alphabet, and let \(\mathcal B_K=\bigoplus _{u\in \Omega _K}M_{D}(\mathbb {C})\) with its Hilbert–Schmidt direct-sum norm. Define
by applying the one-site left virtual map independently to each prefix fiber. Define
by applying the length-\(K\) tail virtual map independently to each final-site fiber. These definitions make sense without any nonemptiness assumption on the virtual dimension or on either configuration set.
When \(D\geq 1\) and \(\rho \in M_{D}(\mathbb {C})\) is positive definite, set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and define the centered virtual residual by
For every \(K\in \mathbb {N}\), the common overlap factors have fiber formulas
and norm bounds
The formulas and bounds remain valid for empty direct sums and incur no factor depending on the number of configurations.
Assume \(D\geq 1\) and that \(\rho \in M_{D}(\mathbb {C})\) is positive definite. The centered residual of Definition 12.3.7.47 factors exactly at the generally nonidentity limiting metric as
and hence
If, in addition, \(A\) is primitive with fixed-point witness \(\rho \), then there are constants \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that, for every \(K\in \mathbb {N}\) and every \(l\geq 1\),
All formulas in this theorem are reconstructed algebra. They are not sourced formulas and assert neither a full projector-defect estimate nor the FNW–Nachtergaele rational coefficient.
Evaluate the two factors on each common direct-sum fiber. The direct-sum norm identity gives the two bounds without a factor counting spectator configurations. Substitute these fiber formulas into the centered residual identity from Theorem 12.3.5.25 to obtain the exact factorization. Submultiplicativity and Theorem 12.3.5.20 give the product bound. Under primitivity, combine the uniform bound for \(J^{\mathrm{tail}}_K\) with the squared geometric Gram estimate and absorb the fixed norm of \(J^{\mathrm{left}}_1\) into the constant.
Let \(S\) be a finite set and let \(G,H\colon V\to V\) be continuous linear endomorphisms of the virtual Euclidean space. Then
Evaluate both sides on each fiber indexed by \(s\in S\).
If the ground-space map \(\Gamma _n^{\mathrm{ES}}\) is injective, then its Gram endomorphism \(\mathcal K_n=(\Gamma _n^{\mathrm{ES}})^\dagger \Gamma _n^{\mathrm{ES}}\) is invertible.
The equality \(\langle X,\mathcal K_nX\rangle =\lVert \Gamma _n^{\mathrm{ES}}X\rVert ^2\) shows that \(\mathcal K_n\) is injective. An injective endomorphism of a finite-dimensional space is bijective and hence invertible.
Assume \(D\geq 1\) and let \(\rho \in M_{D}(\mathbb {C})\) be positive definite. Set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). If \(\Gamma _n^{\mathrm{ES}}\) is injective and \(S_n=\mathcal K_n^{-1}\), then
Both Gram endomorphisms are invertible. Apply the identity \(A^{-1}-B^{-1}=B^{-1}(B-A)A^{-1}\) with \(A=\mathcal K_n\) and \(B=\mathcal K_\infty \).
Suppose that \(\Gamma _K^{\mathrm{ES}}\) is injective and write \(S_K=\mathcal K_K^{-1}\). Then
The direct-sum Gram and inverse Gram cancel fiberwise. The remaining composition is the adjoint of \(\Gamma _{K,L}^{\mathrm{left}}J_L^{\mathrm{left}} =\Gamma _{K+L}^{\mathrm{ES}}\).
If \(\Gamma _L^{\mathrm{ES}}\) is injective, then
The square-root pseudoinverse identity gives the left-hand side as the square root of the norm of the tail boundary inverse Gram. This inverse Gram is the fiberwise lift of \(S_L\), whose norm is at most \(\lVert S_L\rVert \).
If \(\Gamma _K^{\mathrm{ES}}\) is injective, then
The adjoint-side square-root identity reduces the left-hand side to the norm of the left boundary inverse Gram. The direct-sum inverse-Gram formula and the fiberwise norm estimate give the result.
Assume \(D\geq 1\) and let \(\rho \in M_{D}(\mathbb {C})\) be positive definite. Set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). Suppose that \(\Gamma _{l+1}^{\mathrm{ES}}\), \(\Gamma _{K+l}^{\mathrm{ES}}\), and \(\Gamma _{K+l+1}^{\mathrm{ES}}\) are injective, and write
The three correction sandwiches in Theorem 12.3.5.25 are
Under the hypotheses of Definition 12.3.7.55, the three corrections have the exact factorizations
The direct-sum inverse-Gram formula identifies the initial factor with the pseudoinverse factor of \(T\). Theorem 12.3.7.52 replaces the final product in \(Q_1\) and \(Q_2\) by \(C^\dagger \). For \(Q_2\), apply the resolvent identity
Fiberwise subtraction gives the displayed Gram differences. The factorization of \(Q_3\) leaves the left pseudoinverse adjacent to \(L^\dagger \).
Under the hypotheses of Definition 12.3.7.55,
Apply submultiplicativity to the factorizations in Theorem 12.3.7.56. The two spectator pseudoinverse estimates control the exterior factors, and Theorem 12.3.5.20 controls every fiberwise Gram factor without a term depending on \(|\Omega _K|\). The square-root pseudoinverse identity controls \(S_{K+l+1}C^\dagger \).
Let \(A\) be a normalized primitive tensor with nonzero bond dimension and a positive-definite fixed point. For every \(\varepsilon {\gt}0\), all sufficiently large \(L\) and every \(K\geq 0\), \(Q{\gt}0\) satisfy
This is the uniform decay of the individual-block errors used in [ 225 , Section 6 ] .
The geometric FNW estimate bounds the norm by \(c\lambda ^L(1+c\lambda ^L)/(1-c\lambda ^L)\) once \(L\) exceeds the injectivity length and \(c\lambda ^L{\lt}1\). Since \(0{\lt}\lambda {\lt}1\), this coefficient tends to zero. Neither the coefficient nor the lower bound on \(L\) depends on \(K\) or \(Q\).
Let \(A\) be primitive with positive-definite fixed point \(\rho \) and nonzero bond dimension. There is an integer \(p{\gt}0\), measured in sites of \(A\), such that \(A\) is \(p\)-block injective and, for every prefix length \(K\),
Primitivity supplies a positive injective length. Choose \(p\) above it and large enough for the uniform decay estimate with tolerance \(7/16\). Block injectivity propagates to \(p\). Set the suffix length equal to \(p\) in that estimate, leaving the prefix length unrestricted.
Under the same hypotheses, there is an integer \(p{\gt}0\), measured in sites of \(A\), such that \(A\) is \(p\)-block injective and
Apply the prefix-uniform theorem at \(K=p\). The three consecutive regions then each contain exactly \(p\) sites of the input tensor.
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There are an overlap length \(l{\gt}1\), measured in sites of \(A\), and a number \(\epsilon _l\geq 0\) such that \(A\) is \(l\)-block injective, the projector defect is bounded uniformly in the left length \(K\) by \(\epsilon _l\), and
Nachtergaele’s projector-defect identity following condition C3 [ 225 , Eq. (2.4) ] , together with the two-projection estimate, gives the corresponding anticommutator bound for every \(K\). If \(A\) is already a blocked representative, one site of \(A\) is one blocked-chain filtration step.
Apply Theorem 12.9.5 with \(a=1\) and choose one of its sufficiently large lengths \(l\). Since \(\sqrt{l+1}{\gt}0\), the inequality \(\epsilon _l\sqrt{l+1}{\lt}1\) gives \(\epsilon _l{\lt}1/\sqrt{l+1}\). The open-chain projector-defect identity gives the anticommutator statement.
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There are \(l{\gt}1\) and \(\epsilon _l\geq 0\), with \(A\) \(l\)-block injective, such that for every \(K{\gt}0\), after setting \(n=K+l\) and \(n_l=l+1\),
The coefficient satisfies
The length \(l\) and coefficient \(\epsilon _l\) are exactly those chosen by Theorem 12.3.7.61.
The uniform threshold theorem gives
for every \(K{\gt}0\). Rewrite its norm by Theorem 12.3.7.23; the existing strict threshold on \(\epsilon _l\) is unchanged.
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There are \(l{\gt}1\) and \(\epsilon _l\geq 0\), with \(A\) \(l\)-block injective, such that for every final volume \(N\) and every \(n{\lt}N\),
This is the same overlap length and the same coefficient as in Theorem 12.3.7.62. More generally, for any tensor of positive bond dimension that is \(l\)-block injective with \(l{\gt}1\), a uniform physical open-chain bound \(\epsilon \ge 0\) gives the same bound in every final volume, without requiring \(\epsilon {\lt}1/\sqrt{l+1}\).
If \(n{\lt}l\), then \(E_n^{(N)}=0\). If \(n=l\), the endpoint identity gives \(G_{\Lambda _{l+1}}^{(N)}E_l^{(N)}=0\). These cases use only \(\epsilon _l\geq 0\).
Suppose that \(n{\gt}l\), and set \(K=n-l{\gt}0\) and \(r=N-(n+1)\). The prefix and whole-prefix conjugacies identify \(E_n^{(N)}\) with the direct sum of the \((n+1)\)-site martingale difference over the \(r\) right spectators. The interval-projector conjugacy and fiberwise composition then identify the product in (260) with the direct sum of its open-chain counterpart. Isometric conjugacy preserves operator norm, and the fiberwise extension does not increase it. The bound therefore follows from Theorem 12.3.7.62.
Let \(D\geq 1\), let \(l{\gt}0\), suppose that \(A\) is \(l\)-block injective, and let \(N\geq l+1\). Assume \(0\leq \epsilon _l{\lt}1/\sqrt{l+1}\) and, for every \(0\leq n{\lt}N\),
where \(Q_n\) is the ground-space projection of the interval \([n-l,n+1)\) and \(E_n^{(N)}\) is the fixed-volume martingale difference. Then every \(v\in (G_N^{\mathrm{ES}}(A))^\perp \) satisfies
This intermediate MPS specialization assumes C3 on the full range \(0\leq n{\lt}N\), rather than only above the lower threshold in the source statement. It applies the full-range estimate with \(d_{l+1}=\gamma _{l+1}=1\) and retains the coefficient of Theorem 2.1(i) in [ 225 ] . It is not a formulation of the unrestricted source theorem. Theorem 12.3.7.65 supplies the full-range C3 hypothesis from the physical open-chain estimates.
Use the fixed-volume prefix ground-space projections and their differences \(E_n^{(N)}\). Theorem 12.3.4.17 gives \(G_0^{(N)}=I\), while Theorem 12.3.4.16 identifies \(\operatorname {ran}G_N^{(N)}\) with the kernel of the open Hamiltonian. The outside-window commutation theorem gives the locality hypothesis in the martingale estimate. Theorems 12.3.7.9 and 12.3.7.11 give C1 and C2 with \(d_{l+1}=\gamma _{l+1}=1\), while (262) is C3. Positivity of the open Hamiltonian and \(\ker H^{\mathrm{open}}_{N,l+1}(A)=G_N^{\mathrm{ES}}(A)\) identify the final orthogonal complement. Substitution into (143) yields (263).
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There exist \(l{\gt}1\) and \(\epsilon _l\geq 0\) such that \(A\) is \(l\)-block injective,
and, for every \(N\geq l+1\) and every \(v\in (G_N^{\mathrm{ES}}(A))^\perp \),
Theorem 12.3.7.63 supplies \(l{\gt}1\), \(\epsilon _l\), \(l\)-block injectivity, and the C3 estimate for every final volume \(N\) and every \(n{\lt}N\). Apply Theorem 12.3.7.64 for each \(N\geq l+1\). The value of \(\epsilon _l\) is unchanged, so the coefficient is exactly the specialization
Let \(A\) be an MPS tensor with nonzero physical dimension, let \(p\) be any blocking length, and set \(B=\operatorname {block}_p(A)\). Then
If \(B\) is injective, then the bound on either side
gives the adjacent three-site anticommutator inequality
The same quadratic-form inequality holds for every forward or backward adjacent pair of cyclic range-two terms on every chain of length at least three, including pairs crossing the periodic cut.
Let \(A\) be a primitive MPS tensor with nonzero physical and bond dimensions, and suppose that its fixed point is positive definite. There is a blocking length \(p{\gt}0\) such that \(A\) is \(p\)-block injective and, with \(B=\operatorname {block}_p(A)\), for every \(N\geq 4\) and every \(v\in (\ker H_{N,2}(B))^\perp \),
Choose \(p\) so that the three-block projector defect is at most \(7/16\). Lemma 12.3.7.66 gives the same coefficient for every overlapping cyclic pair. For range two each row has at most two such pairs, so the martingale estimate gives \(1-2(7/16)=1/8\).
Let \(P\) and \(Q\) be orthogonal projections on a real or complex Hilbert space. For every vector \(v\),
Positivity of the inner product gives
where the equality expands the inner product using symmetry and idempotence \(P^\dagger =P=P^2\), \(Q^\dagger =Q=Q^2\). Rearranging (272) proves the claim.
Let \(H\) be a positive operator on a finite-dimensional complex Hilbert space and let \(\gamma \geq 0\). If
then every vector \(v\) satisfies
Put \(w=v-P_{\ker H}v\). Then \(w\perp \ker H\), \(Hw=Hv\), and symmetry gives \(\langle Hv,v\rangle =\langle Hv,w\rangle \). Hence
Let \(G\) act on a finite active configuration space. If \(G^2\geq \gamma G\) as a quadratic form, the same inequality holds when \(G\) acts independently over any finite right spectator coordinate. In particular, positivity and a norm gap \(\gamma \) for \(G\) imply this fiberwise quadratic-form inequality.
Write \(v_s\) for the active vector at spectator value \(s\). Both quadratic forms split into finite sums over \(s\), and the active inequality gives
The final statement follows from Lemma 12.3.7.69.
Let \(h_i\), with \(i\in \mathbb Z/N\mathbb Z\), be orthogonal projections. Define
Then, for every vector \(v\),
For fixed offsets \(a,b\in \{ 0,\ldots ,m-1\} \), translation by \(a\) gives
The case \(a=b\) contributes \(mH\), while each of the differences \(b-a=q\) and \(b-a=-q\) contributes \((m-q)Q_q\). Summing (284) over \(a\) and \(b\) gives (281). Taking the quadratic form of that operator identity at \(v\) turns each product \(h_ih_{i\pm q}\) into the pair \(\operatorname {Re}\langle h_iv,h_{i\pm q}v\rangle \), giving (282). Translation invariance also gives
For \(Q_q\) as in Theorem 12.3.7.71,
Suppose that \(h_i\) and \(h_{i+d}\) commute whenever \(R\leq d\leq N-R\). Then the cross term is nonnegative for these offsets. If \(1\leq m\), \(R\leq m\), and \(2m\leq N\), it follows that
Translation of the reversed leg gives
Summing Theorem 12.3.7.68 over \(i\) and using (288) proves the upper bound. If \(h_i\) and \(h_{i+d}\) commute, their symmetry and idempotence give
Partitioning the nonzero cyclic offsets into \(1\leq q{\lt}m\), their reflections \(N-q\), and \(m\leq k\leq N-m\) gives
The last sum in (290) is nonnegative by (289), which proves the lower bound.
Let \(1\leq R\leq m\), \(2m\leq N\), and let \(h_i\) be cyclically indexed orthogonal projections such that \(h_i\) commutes with \(h_{i+d}\) whenever \(R\leq d\leq N-R\). Suppose that, for every \(s\) and \(v\),
If \((R-1)^2{\lt}m\gamma \), then, with
one has
The coefficient for general \(R\) is derived here and is not attributed to the cited papers.
Put \(c=m-R+1\). Summing the window gaps and applying Theorem 12.3.7.71 gives
Decompose \(m-q=c+(R-1-q)\) and expand (282): the weights \(q\leq R-2\) carry the coefficient \(c\) plus the positive part \(R-1-q\), while for \(q\geq R\) the coefficient \(R-1-q\) is nonpositive and the cross term is nonnegative, so those summands are bounded by \(c\, C_q(v)\). For the positive parts, the anticommutator bound and the triangular identity
combine with the decomposition
so that
The cyclic cross-term estimate therefore gives
Rearranging and dividing by \(c{\gt}0\) proves the claim.
Under the hypotheses of Theorem 12.3.7.73 with \(R=2\), if \(1{\lt}m\gamma \), then
Equivalently, the open-window threshold is \(\gamma {\gt}1/m\). This is Knabe’s nearest-neighbor threshold [ 211 ] , quoted in [ 228 , lines 1483–1489 ] and [ 17 , lines 2194–2197 ] .
Substitute \(R=2\) into the coefficient of Theorem 12.3.7.73.
For an \(N\)-site periodic chain with \(N{\gt}0\), index the range-\(R\) local interactions by \(\mathbb Z/N\mathbb Z\). If \(s\in \mathbb Z/N\mathbb Z\) and \(i\in \{ 0,\ldots ,N-1\} \) is its standard representative, write
The sum over the cyclic coordinates is the periodic parent Hamiltonian:
Reindex the standard sum over \(\{ 0,\ldots ,N-1\} \) by its canonical bijection with \(\mathbb Z/N\mathbb Z\). Each summand is unchanged.
Let \(R\leq e\) and \(e+R\leq N\). Addition by \(e\) in \(\mathbb Z/N\mathbb Z\) moves the standard representative forward by \(e\) sites. The two length-\(R\) cyclic windows beginning at \(s\) and \(s+e\) are disjoint. Consequently,
Measure every site from the first window’s initial site in the forward cyclic order. Membership in both windows would give an offset smaller than \(R\) and also an offset of the form \(e+b\) with \(b{\lt}R\). The inequalities \(R\leq e\) and \(e+R\leq N\) exclude this possibility. Local interactions on disjoint cyclic windows commute, which gives (301).
Let \(W\leq N\) and fix a site \(i\). Splitting the cyclically ordered sites into the first \(W\) sites and their complement gives
If \((\sigma ,\tau )\) is an active configuration and a complementary configuration, then their inverse image has value \(\sigma _r\) at the site \(i+r\) for \(0\leq r{\lt}W\), and value \(\tau _r\) at the site \(i+W+r\) for \(0\leq r{\lt}N-W\). The isometry and its inverse act by composition with this configuration bijection.
Suppose that \(W\leq N\) and \(q+R\leq W\). Under the isometry \(U_i\) in (303), the range-\(R\) interaction beginning at the cyclic site \(i+q\) becomes the corresponding nonwrapping interaction on the active \(W\)-site interval, acting independently on the complementary configuration:
The condition \(q+R\leq W\) places the whole interaction window inside the active block. Joining an active configuration to a complementary one therefore preserves both the extracted length-\(R\) word and every configuration obtained by replacing that word. The local projector acts only on the active coordinates, which proves (304) on each complementary configuration.
Let \(1\leq R\leq m\) and \(2m\leq N\), and put
For every \(s\in \mathbb Z/N\mathbb Z\), let \(A_s=\sum _{q=0}^{m-1}h_{s+q}^{(N,R)}\). The \(m\) interaction windows occupy exactly the \(W\) cyclic sites beginning at \(s\). Under the active block isometry \(U_s\),
Let \(1\leq R\leq m\), put \(W=m+R-1\), and suppose that \(H^{\mathrm{open}}_{W,R}(A)\) has norm gap \(\gamma \). Thus, for every \(u\in (\ker H^{\mathrm{open}}_{W,R}(A))^\perp \),
If \((R-1)^2{\lt}m\gamma \), define
Then \(\delta {\gt}0\), and for every \(N\geq 2m\) and every \(v\in (\ker H_{N,R}^{\mathrm{ES}}(A))^\perp \),
The coefficient in (308) is the TNLean finite-range derivation documented in docs/paper-gaps/knabe88_finite_range_coefficient.tex. It is not a theorem of the MPS sources cited for the gap definition and the nearest-neighbor Knabe principle.
The norm gap (307) and positivity give \((H^{\mathrm{open}}_{W,R})^2\geq \gamma H^{\mathrm{open}}_{W,R}\) as a quadratic form. Lemma 12.3.7.70 gives the same inequality after adjoining the \(N-W\) complementary sites, and Theorem 12.3.7.80 gives it for every cyclic sum \(A_s\). The oriented-separation theorem verifies the commutation hypothesis in Theorem 12.3.7.73. That theorem gives
as a quadratic form, using the total-sum identity (300). The inequality \((R-1)^2{\lt}m\gamma \) makes \(\delta \) positive, and the finite-dimensional spectral criterion yields (309).
Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There exist \(l{\gt}1\), \(\epsilon _l\geq 0\), an integer \(m\geq l+1\), and \(\delta {\gt}0\) such that \(A\) is \(l\)-block injective and
Set
The integer \(m\) may be chosen so that
and
For every \(N\geq 2m\) and every \(v\in (\ker H_{N,l+1}^{\mathrm{ES}}(A))^\perp \),
The existence of a uniform MPS parent-Hamiltonian gap and the nearest-neighbor Knabe principle are stated in [ 228 , lines 1460–1500 ] and [ 17 , lines 2184–2188,2194–2197 ] . Those passages do not state the arbitrary finite-range coefficient (316).
Theorem 12.3.7.65 gives \(l\), \(\epsilon _l\), and the norm gap \(\gamma _{\mathrm{open}}\) on every open volume at least \(l+1\). Since \(\gamma _{\mathrm{open}}{\gt}0\), choose \(m\geq l+1\) with \(l^2{\lt}m\gamma _{\mathrm{open}}\). At the active volume \(W=m+l\), block injectivity identifies the open ground space with the kernel of the open Hamiltonian, so (265) is exactly the hypothesis (307) with \(R=l+1\). Apply Theorem 12.3.7.81. Substituting \(R-1=l\) and \(m-R+1=m-l\) gives (316) and (317).
12.4 Overlaps of distinct block ground spaces
Let \(A\) and \(B\) have bond dimensions \(D_A\) and \(D_B\), respectively, and let \(\Gamma _n^{\mathrm{ES}}(A)\) and \(\Gamma _n^{\mathrm{ES}}(B)\) be their Hilbert-space boundary maps, writing \(E_{ab}\) also for the coordinate vector of a matrix unit. Let \(F_{BA}^{\mathrm{rect}}\) be the rectangular mixed transfer operator on \(M_{D_B\times D_A}(\mathbb {C})\). For matrix units \(E_{ab}\) and \(E_{ce}\),
Thus \(\Gamma _n^{\mathrm{ES}}(A)^*\Gamma _n^{\mathrm{ES}}(B)\) is the rectangular reshuffling of \((F_{BA}^{\mathrm{rect}})^n\). This is the boundary-coordinate form of equations ip and P12 in the proof of the lemma disjoint in [ 225 ] .
Expanding the trace boundary maps gives
The word expansion of \((F_{BA}^{\mathrm{rect}})^n\) gives the assertion.
If \((F_{BA}^{\mathrm{rect}})^n(X)\to 0\) for every boundary matrix \(X\), then \(\lVert \Gamma _n^{\mathrm{ES}}(A)^*\Gamma _n^{\mathrm{ES}}(B)\rVert \to 0\). In particular this holds when the spectral radius of \(F_{BA}^{\mathrm{rect}}\) is less than one.
Theorem 12.4.1 gives convergence of every entry of the mixed Gram operator. Its domain and codomain have fixed finite dimensions, so entrywise convergence implies convergence in operator norm. A spectral radius less than one gives decay of the powers of \(F_{BA}^{\mathrm{rect}}\), as in equation limP12 in the proof of the lemma disjoint in [ 225 ] .
Suppose that \(\Gamma _n^{\mathrm{ES}}(A)\) and \(\Gamma _n^{\mathrm{ES}}(B)\) are injective, and put \(\mathcal K_n(A)=\Gamma _n^{\mathrm{ES}}(A)^*\Gamma _n^{\mathrm{ES}}(A)\) and \(\mathcal K_n(B)=\Gamma _n^{\mathrm{ES}}(B)^*\Gamma _n^{\mathrm{ES}}(B)\). For vectors \(x\in \operatorname {ran}\Gamma _n^{\mathrm{ES}}(A)\) and \(y\in \operatorname {ran}\Gamma _n^{\mathrm{ES}}(B)\),
Write \(x=\Gamma _n^{\mathrm{ES}}(A)u\) and \(y=\Gamma _n^{\mathrm{ES}}(B)v\). The left inverse \(\mathcal K_n(A)^{-1}\Gamma _n^{\mathrm{ES}}(A)^*\) has norm \(\sqrt{\lVert \mathcal K_n(A)^{-1}\rVert }\), so \(\lVert u\rVert \leq \sqrt{\lVert \mathcal K_n(A)^{-1}\rVert }\lVert x\rVert \); the analogous estimate holds for \(v\). Apply these bounds to \(|\langle u,\Gamma _n^{\mathrm{ES}}(A)^*\Gamma _n^{\mathrm{ES}}(B)v\rangle |\). This makes explicit the normalization step using equation C1C2 in [ 225 ] .
Let \(A\) and \(B\) be trace-preserving tensors with positive-definite fixed matrices \(\rho \) and \(\sigma \). Suppose their complementary transfer maps have spectral radius less than one, and that \(\rho _{\operatorname{spec}}(F_{BA}^{\mathrm{rect}}){\lt}1\). For every \(\varepsilon {\gt}0\), there is \(n_0\) such that for every \(n\geq n_0\), \(x\in G_n(A)\), and \(y\in G_n(B)\),
This is the bound on inner products of ground-space vectors obtained from the spectral-radius estimate in the proof of the lemma disjoint in [ 225 ] .
By Theorem 11.4.23, the self-Gram operators \(\mathcal K_n(A)\) and \(\mathcal K_n(B)\) converge to invertible limits \(\mathcal K_\infty (A)\) and \(\mathcal K_\infty (B)\) determined by \(\rho \) and \(\sigma \), and their inverses converge to \(\mathcal K_\infty (A)^{-1}\) and \(\mathcal K_\infty (B)^{-1}\). By Theorem 11.4.22, \(\mathcal K_n(A)\) and \(\mathcal K_n(B)\) are eventually invertible, and hence \(\Gamma _n^{\mathrm{ES}}(A)\) and \(\Gamma _n^{\mathrm{ES}}(B)\) are injective, for all sufficiently large \(n\). By Theorem 12.4.2, the coefficient in Theorem 12.4.3 satisfies
Choose \(n_0\) with \(c(n)\leq \varepsilon \) for \(n\geq n_0\); then Theorem 12.4.3 gives the bound, uniformly in \(x\) and \(y\).
Let \(A\) and \(B\) be trace-preserving tensors with positive-definite fixed matrices and complementary transfer spectral radii less than one. Suppose that, if their bond dimensions agree, they are not gauge-phase equivalent. Then, for every \(\varepsilon {\gt}0\) and all sufficiently large \(n\), every \(x\in G_n(A)\) and \(y\in G_n(B)\) satisfy
This is the normalized tensor form of equation limitepsilonm in [ 225 ] .
The positive-definite fixed matrices and complementary gaps imply irreducibility by Lemma C.5.7. For equal bond dimensions, inequivalence and Theorem 5.6.2 give \(\rho _{\operatorname{spec}}(F_{BA}^{\mathrm{rect}})=\rho _{\operatorname{spec}}(F_{BA}){\lt}1\). For unequal bond dimensions, a peripheral eigenvector of \(F_{BA}^{\mathrm{rect}}\) would force equality of the bond dimensions, so Theorem 5.6.4 gives
Apply Theorem 12.4.4.
12.5 Overlap with a sum of subspaces
Let \(v_1,\ldots ,v_m\) lie in a complex inner product space, and let \(B\in M_m(\mathbb {R})\) have zero diagonal. Suppose that \(|\langle v_i,v_j\rangle |\leq B_{ij}\lVert v_i\rVert \lVert v_j\rVert \) whenever \(i\neq j\). With the Euclidean operator norm on \(B\),
This is the lower Gram estimate in the proof of [ 225 , Lemma 9 ] .
Put \(w_i=\lVert v_i\rVert \). Expanding the squared norm gives
Rearranging proves the assertion.
Let \(U,V_1,\ldots ,V_m\) be subspaces of a complex inner product space. Let \(a\in \mathbb {R}^m\) and \(B\in M_m(\mathbb {R})\), where \(B\) has zero diagonal and \(\lVert B\rVert {\lt}1\). Suppose that \(|\langle u,v\rangle |\leq B_{ij}\lVert u\rVert \lVert v\rVert \) for \(i\neq j\), \(u\in V_i\), and \(v\in V_j\). Suppose also that \(|\langle x,u\rangle |\leq a_i\lVert x\rVert \lVert u\rVert \) for every \(i\), \(x\in U\), and \(u\in V_i\). Then every \(x\in U\) and \(y\in V_1+\cdots +V_m\) satisfy
This is [ 225 , Lemma 9 ] , expressed as a bound on inner products.
Write \(y=\sum _i v_i\) with \(v_i\in V_i\). The triangle inequality and Cauchy–Schwarz give
Apply Theorem 12.5.1 to the final factor.
Let \(U,V_1,\ldots ,V_m\) be subspaces of a complex inner product space, and let \(\varepsilon \geq 0\) satisfy \(\varepsilon m\leq 1\). Suppose that every pair of vectors in distinct \(V_i\) satisfies \(|\langle u,v\rangle |\leq \varepsilon \lVert u\rVert \lVert v\rVert \), and that the same inequality holds for vectors in \(U\) and \(V_i\), for every \(i\). Set
Then \(\eta \leq 1\), and every \(x\in U\) and \(y\in V_1+\cdots +V_m\) satisfy \(|\langle x,y\rangle |\leq \eta \lVert x\rVert \lVert y\rVert \). This is the uniform bound in [ 225 , Lemma 9 ] .
For \(y=\sum _i v_i\), the off-diagonal terms satisfy
Take \(a_i=\varepsilon \) and \(B_{ij}=\varepsilon (1-\delta _{ij})\). For \(w\in \mathbb {R}^m\) write \(|w|\) for the vector with entries \(|w_i|\). Since \(B\) is symmetric with nonnegative entries, the same computation applied to \(|w|\) gives
and hence \(\lVert B\rVert =\sup _{\lVert w\rVert =1}|\langle w,Bw\rangle | \leq \varepsilon (m-1)\leq 1-\varepsilon \), which is less than one when \(\varepsilon {\gt}0\) and vanishes when \(\varepsilon =0\). With \(\lVert a\rVert =\varepsilon \sqrt m\), Theorem 12.5.2 gives
Finally, \(\varepsilon ^2m\leq 1-\varepsilon (m-1)\) follows from \(\varepsilon m\leq 1\), and gives \(\eta \leq 1\).
12.6 Comparison of sums of orthogonal projectors
Let \((v_i)\) and \((w_i)\) be finite families in a complex inner product space. Let \(B\) be a real matrix with zero diagonal and \(\beta =\lVert B\rVert {\lt}1\). Suppose that, for \(i\ne j\), each of \(|\langle v_i,v_j\rangle |\), \(|\langle w_i,w_j\rangle |\), and \(|\langle v_i,w_j\rangle |\) is bounded by \(B_{ij}\) times the product of the corresponding vector norms. Put \(\delta =\beta /(1-\beta )\). Then
This is the pairing estimate in the proof of [ 225 , Lemma 6.1 ] .
Write \(a_i=\lVert v_i\rVert \) and \(b_i=\lVert w_i\rVert \). The left-hand side is at most \(\langle a,Bb\rangle \leq \beta \lVert a\rVert \lVert b\rVert \). The lower Gram estimate gives \(\lVert a\rVert \leq \lVert \sum _i v_i\rVert /\sqrt{1-\beta }\), and likewise for \(b\).
Let \((V_i)\) be a finite family of subspaces of a finite-dimensional complex inner product space. Let \(B\) have zero diagonal and \(\beta =\lVert B\rVert {\lt}1/2\), and suppose that \(|\langle u,v\rangle |\leq B_{ij}\lVert u\rVert \lVert v\rVert \) for \(i\ne j\), \(u\in V_i\), and \(v\in V_j\). Write \(P_i\) for the orthogonal projector onto \(V_i\), \(P\) for the projector onto \(\sum _i V_i\), \(X=\sum _i P_i\), and \(\delta =\beta /(1-\beta )\). Then
This is the finite-dimensional projector estimate underlying [ 225 , Lemma 6.1(i) ] .
For \(u=\sum _j v_j\) with \(v_j\in V_j\), self-adjointness gives
The pairing estimate bounds its modulus by \(\delta \lVert Xz\rVert \lVert u\rVert \). Since \(XP=X\), it follows that \(\lVert X-P\rVert \leq \delta \lVert X\rVert \). Combine this with \(\lVert X\rVert \leq \lVert X-P\rVert +1\) and \(\delta {\lt}1\). For the second bound, use \(\lVert P\rVert \leq 1\) and the first bound:
Let \((U_i)\) and \((V_i)\) be finite families of subspaces of a finite-dimensional complex inner product space. Let \(B\) have zero diagonal and \(\beta =\lVert B\rVert {\lt}1\). Suppose that, for \(i\ne j\), \(|\langle u,v\rangle |\leq B_{ij}\lVert u\rVert \lVert v\rVert \) whenever \((u,v)\) lies in \(U_i\times U_j\), \(V_i\times V_j\), or \(U_i\times V_j\). Write \(P_i,Q_i\) for their orthogonal projectors and put \(X_U=\sum _i P_i\), \(X_V=\sum _i Q_i\), \(\delta =\beta /(1-\beta )\). Then
Pair the operator on the left with unit vectors \(z,x\). Its matrix coefficient is \(\sum _{i\ne j}\langle P_i z,Q_j x\rangle \). Apply the off-diagonal pairing estimate and take the supremum over \(z,x\).
Under the hypotheses of Theorem 12.6.3, assume \(\beta {\lt}1/2\). Let \(W_i\subseteq U_i\), and let \(R_i\) be the orthogonal projector onto \(W_i\). Let \(P,Q,R\) project onto \(\sum _i U_i\), \(\sum _i V_i\), and \(\sum _i W_i\), respectively. Then
This is the abstract projector comparison used in [ 225 , Lemma 6.1(ii) ] .
Put \(X_W=\sum _i R_i\), \(Z=\sum _i P_iQ_i\), \(a=\delta /(1-\delta )\), and \(q=1/(1-\delta )\). The triangle inequality, applied through \(X_W\), \(Z\), and \(X_U X_V\), gives
Here \(\lVert X_U X_V-PQ\rVert \leq aq+a\) follows by expanding \((X_U-P)X_V+P(X_V-Q)\), and \(a\leq aq\) since \(q\geq 1\).
12.7 Overlap bounds with spectator sites
Throughout this section, local ground spaces are subspaces of the Hilbert spaces of configurations, so \(G_L(A)\) stands for \(G_L^{\mathrm{ES}}(A)=\iota _L(G_L(A))\) of Definition 11.3.1.
Let \(A,B\) be tensors with the same physical dimension and arbitrary bond dimensions. Suppose \(\varepsilon \geq 0\) and \(|\langle a,b\rangle |\leq \varepsilon \lVert a\rVert \lVert b\rVert \) for every \(a\in G_L(A)\) and \(b\in G_L(B)\). Let \(x,y\) be vectors on three successive intervals of lengths \(K,L,Q\). For each prefix configuration \(u\) and suffix configuration \(v\), define \(x_{u,v}(w)=x(u,w,v)\) and \(y_{u,v}(w)=y(u,w,v)\). If \(x_{u,v}\in G_L(A)\) and \(y_{u,v}\in G_L(B)\) for all \(u,v\), then
No positivity assumption on \(K,L,Q\) is required. This is the finite-sum estimate used to extend the overlap bounds in [ 225 , Lemma 6.1(ii) ] .
Decomposing along the prefix and suffix configurations gives
The triangle inequality, the overlap bound, and Cauchy–Schwarz for the sequences \((\lVert x_{u,v}\rVert )\) and \((\lVert y_{u,v}\rVert )\) give
Assume the ground-space overlap bound of Theorem 12.7.1. Let \(U_A\) be the range of \(\Gamma ^{\mathrm{left}}_{K+L,Q}(A)\) and \(V_A\) the range of the reassociated tail boundary map \(\widetilde\Gamma ^{\mathrm{tail}}_{K;L,Q}(A)\), both in the physical space on \(K+L+Q\) sites. Define \(U_B,V_B\) in the same way. Then \(|\langle x,y\rangle |\leq \varepsilon \lVert x\rVert \lVert y\rVert \) for vectors in any of the pairs of ranges \(U_A\times U_B\), \(V_A\times V_B\), and \(U_A\times V_B\).
For a left boundary vector with boundary family \((Z_v)\), fixing \(u,v\) leaves the middle vector \(\Gamma _L(Z_vA^u)\). For a right boundary vector with boundary family \((Y_u)\), the corresponding middle vector is \(\Gamma _L(A^vY_u)\). Both belong to the appropriate length-\(L\) ground space, so the common-interval estimate applies to each of the three pairs.
Let \((A_i)\) be a finite family of tensors with the same physical dimension, with bond dimensions allowed to depend on \(i\). Let \(B\) be a nonnegative real matrix with zero diagonal and \(\beta =\lVert B\rVert {\lt}1/2\). Suppose that for \(i\ne j\), \(x\in G_L(A_i)\), and \(y\in G_L(A_j)\), \(|\langle x,y\rangle |\leq B_{ij}\lVert x\rVert \lVert y\rVert \). In the physical space on \(K+L+Q\) sites, put
Let \(P_i,Q_i,R_i\) be the orthogonal projectors onto these spaces, and let \(P,Q,R\) project onto their respective sums. With \(\delta =\beta /(1-\beta )\),
This is the physical interval comparison in the proof of [ 225 , Lemma 6.1(ii) ] , before inserting the individual block error bounds.
The boundary-map factorization gives \(W_i\subseteq U_i\). The extended-boundary overlap estimate supplies all three bounds for \(U_i\times U_j\), \(V_i\times V_j\), and \(U_i\times V_j\) with the same matrix \(B\). Apply the joint-projector comparison.
Let \((A_i)_{i\in I}\) be a finite family of tensors, and use the interval spaces and projectors of Theorem 12.7.3. Suppose that, for each \(\varepsilon {\gt}0\) and all sufficiently large \(L\), the following two bounds hold. For distinct \(i,j\), \(x\in G_L(A_i)\), and \(y\in G_L(A_j)\), \(|\langle x,y\rangle |\leq \varepsilon \lVert x\rVert \lVert y\rVert \). For every \(i\), \(K\geq 0\), and \(Q{\gt}0\), \(\lVert Q_iP_i-R_i\rVert \leq \varepsilon \). Then, for every \(\eta {\gt}0\), all sufficiently large \(L\), and every \(K\geq 0\) and \(Q{\gt}0\),
Put \(r=|I|\) and let \(C\) have zero diagonal and all off-diagonal entries equal to one. As \(\varepsilon \downarrow 0\), the quantities \(\beta =\lVert \varepsilon C\rVert \), \(\delta =\beta /(1-\beta )\), and \(4\delta /(1-\delta )^2\) tend to zero. Choose \(\varepsilon {\gt}0\) so that \(\beta {\lt}1/2\) and the last quantity is less than \(\eta /(r+1)\). For all sufficiently large \(L\), the overlap bound holds with \(B=\varepsilon C\), and every individual projector error is at most \(\eta /(r+1)\). Since \(P_i\), \(Q_i\), and \(R_i\) are self-adjoint, \((R_i-P_iQ_i)^*=R_i-Q_iP_i\), and hence
The interval comparison therefore gives
Let \((A_i)_{i\in I}\) be a finite family of normalized primitive tensors with positive bond dimensions and positive-definite fixed points. Suppose that, if the bond dimensions of two distinct members agree, they are not related by a gauge transformation and a phase. Use the interval projectors of Theorem 12.7.3. For every \(\eta {\gt}0\), all sufficiently large \(L\), and every \(K\geq 0\) and \(Q{\gt}0\),
This is the uniform decay consequence of [ 225 , Lemma 6.1(ii) ] for a positive right outer interval.
Inequivalence of distinct blocks gives, for every \(\varepsilon {\gt}0\), a length \(L_1\) such that for all \(L\ge L_1\), all distinct \(i,j\), and all \(x\in G_L(A_i)\), \(y\in G_L(A_j)\),
Primitivity gives, for every \(\varepsilon {\gt}0\), a length \(L_2\) such that for all \(L\ge L_2\), every block \(i\), and every \(K\ge 0\) and \(Q{\gt}0\),
The family is finite, so both bounds hold at the common threshold
Apply Theorem 12.7.4.
Let \(B=\bigoplus _{j=1}^r\mu _jA_j\) with \(\mu _j\ne 0\) for every \(j\). For every \(K,L\geq 0\), the extended interval spaces satisfy
For every \(K,L,Q\geq 0\), with the reassociation \(\mathscr A_{K,L,Q}\) of Definition 12.3.7.28, the right interval space of length \(L+Q\) satisfies
Let \(\Omega _L\) be the set of physical words of length \(L\). For subspaces \(S_1,\dots ,S_r\) of a vector space, the finite coordinate identity
holds: the right side is contained in the left, and a family \((\psi _\tau )_\tau \) with \(\psi _\tau =\sum _j\psi _{\tau ,j}\) and \(\psi _{\tau ,j}\in S_j\) is the sum over \(j\) of the families \((\psi _{\tau ,j})_\tau \). Let \(T_{K,L}\) be the coordinate isomorphism sending \((\psi _\tau )_{\tau \in \Omega _L}\) to the vector with coefficient \(\psi _\tau (\sigma )\) at \((\sigma ,\tau )\in \Omega _K\times \Omega _L\). Then \(G_K(C)\otimes \mathcal H_L=T_{K,L}\prod _{\tau \in \Omega _L}G_K(C)\) for every tensor \(C\). The local identity \(G_K(B)=\sum _jG_K(A_j)\) and the finite coordinate identity give
since linear maps preserve sums of subspaces. The same argument, with the free sites on the left and coordinates indexed by \(\Omega _K\), gives the second identity. Applying the linear map \(\mathscr A_{K,L,Q}\) to the second identity at lengths \(K\) and \(L+Q\) gives the third.
Let \(B=\bigoplus _{j=1}^r\mu _jA_j\), where every \(\mu _j\) is nonzero and the \(A_j\) are pairwise gauge-phase inequivalent normalized primitive tensors with positive bond dimensions and positive-definite fixed points. Write \(P_I(B)\) for the orthogonal projector onto the ground space on the interval \(I\), extended by the identity on the remaining sites. For every \(\eta {\gt}0\), all sufficiently large \(L\), and every \(K\geq 0\) and \(Q{\gt}0\),
For each block put
and let \(P,S,R\) be the orthogonal projectors onto \(\sum _jU_j\), \(\sum _jV_j\), and \(\sum _jW_j\). These are the projectors written \(P,Q,R\) in Theorem 12.7.3; the middle one is renamed here because \(Q\) denotes a length. The local identity at length \(K+L+Q\), the first identity of Theorem 12.7.6 at lengths \(K+L\) and \(Q\), and its reassociated identity at lengths \(K,L,Q\) give
Hence the left side of the asserted bound is \(\lVert R-PS\rVert \), and Theorem 12.7.5 with the prescribed \(\eta \) bounds it by \(\eta \) for all sufficiently large \(L\), every \(K\geq 0\), and every \(Q{\gt}0\).
12.8 The normalized primitive gauge of a normal tensor
A normal tensor need not be trace-preserving or have spectral radius one. Standard treatments normalize the tensor by scaling the spectral radius of the transfer map to one and applying a gauge transformation by the square root of a positive fixed point [ 225 , 228 ] . This gauge reduction extends the finite-range spectral gap to arbitrary normal tensors.
Let \(\zeta \neq 0\) and let \(A\) be a tensor. For every \(N\) the length-\(N\) word spans of \(A\) and of \(\zeta A\) agree, so \(\zeta A\) is \(N\)-block injective exactly when \(A\) is, and \(\zeta A\) is normal exactly when \(A\) is.
Every length-\(N\) word of \(\zeta A\) is the corresponding word of \(A\) multiplied by \(\zeta ^{N}\), and \(\zeta ^{N}\) is invertible, so each span contains the other.
Let \(D{\gt}0\) and let \(A\) be a normal tensor. Then \(A^i\neq 0\) for at least one \(i\).
If every \(A^i\) vanished, then every word of positive length would vanish, so the span of the length-\(N\) words would be \(\{ 0\} \) for the length \(N{\gt}0\) at which \(A\) is block injective. That span is \(M_{D}(\mathbb {C})\), which contains \(\mathbb {1}\neq 0\) because \(D{\gt}0\).
Let \(X\) be invertible and let \(E\) and \(Y\) be \(D\times D\) matrices. Then
Cyclicity of the trace moves the leading factor \(X\) to the end of the product.
Let \(A\) and \(B\) be gauge equivalent. For every \(L\) one has \(G_L(A)=G_L(B)\).
If \(B^i=XA^iX^{-1}\), then \(B^\sigma =XA^\sigma X^{-1}\) for every length-\(L\) configuration \(\sigma \) by Lemma 2.4.18, so by Lemma 12.8.3 the boundary parametrization of \(B\) at \(Y\) agrees with that of \(A\) at \(X^{-1}YX\). This gives \(G_L(B)\subseteq G_L(A)\), and the symmetric argument applied to \(A^i=X^{-1}B^iX\) gives the reverse inclusion.
Let \(\zeta \neq 0\). For every \(L\) one has \(G_L(\zeta A)=G_L(A)\).
The boundary parametrization of \(\zeta A\) at length \(L\) sends \(X\) to the value of the boundary parametrization of \(A\) at \(\zeta ^{L}X\), and multiplication by \(\zeta ^{L}\) is a bijection of \(M_{D}(\mathbb {C})\).
Let \(A\) and \(B\) have the same bond dimension and satisfy \(G_L(A)=G_L(B)\). Then their canonical parent interactions at length \(L\) agree, hence so do all their translated local terms and their parent Hamiltonians \(H_{N,L}(A)=H_{N,L}(B)\) and \(H_{N,L}^{\mathrm{ES}}(A) =H_{N,L}^{\mathrm{ES}}(B)\) on every periodic chain.
The parent interaction at length \(L\) is the orthogonal projector onto \(G_L(A)^\perp \), so it depends on \(G_L(A)\) alone. Every translated local term is built from that single projector, and the parent Hamiltonian is their sum.
Normality is eventual full Kraus rank (Remark 6.1.1.12), so \(S_N(A)=M_{D}(\mathbb {C})\) for some \(N{\gt}0\). Applying the length-\(N\) words to a nonzero \(\varphi \) then spans \(\mathbb {C}^D\), which is the spreading condition. A subspace invariant under all matrices of \(A\) and containing a nonzero vector therefore contains the span of the images of that vector, namely all of \(\mathbb {C}^D\); so no invariant projection is nontrivial.
Let \(D{\gt}0\) and let \(A\) be a normal tensor. There exist a scalar \(\zeta \neq 0\), a tensor \(B\), and a matrix \(\rho {\gt}0\) such that \(\zeta A\) and \(B\) are gauge equivalent, the pair \((B,\rho )\) is primitive, and
for every chain length \(N\) and every configuration \(\sigma \). The local ground spaces, the parent interactions, and the periodic chain ground spaces of \(A\) and \(B\) also coincide:
for all \(L\) and \(N\), where \(h_L\) is the canonical parent interaction and \(\mathcal G_{N,L}\) the periodic chain ground space.
This is the normalization taken without loss of generality in [ 225 , equations (3.1)–(3.2b), lines 1394–1435 ] and at the start of the proof of [ 228 , Theorem 4, lines 765–770 ] . The gauge here is built from the Perron eigenvector of the adjoint transfer map, giving the trace-preserving orientation. That orientation is what conditions 1 and 2 of [ 228 , Theorem 4, lines 752–758 ] , \(\sum _iA^i(A^i)^\dagger =\mathbb {1}\) and \(\sum _i(A^i)^\dagger \Lambda A^i=\Lambda \) with \(\Lambda {\gt}0\), become after the further gauge by \(\Lambda ^{1/2}\): condition 2 turns into \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\) and condition 1 into the statement that \(\Lambda \) is a positive definite fixed point of the transfer map of \(B\).
Theorem 12.8.7 makes \(A\) an irreducible tensor, with no normalization imposed on \(A\), and Theorem 12.8.2 supplies the nonzero matrix that the Perron–Frobenius eigenvector requires. Theorem 4.5.3 therefore returns a positive definite \(\sigma \) and an eigenvalue \(r{\gt}0\) of the adjoint transfer map. Set \(\zeta =r^{-1/2}\) and \(B^i=\zeta \, \sigma ^{1/2}A^i\sigma ^{-1/2}\); then \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\), and \(\zeta A\) and \(B\) are gauge equivalent. Rescaling preserves normality by Theorem 12.8.1 and gauging preserves it by Theorem 2.5.7, so \(B\) is normal. Since \(B\) is trace preserving, Theorem 6.1.1.16 makes \(B\) strongly irreducible, and Theorem 6.7.4.2 then returns a positive definite \(\rho \) with \((B,\rho )\) primitive. Gauge equivalence leaves the matrix product vectors unchanged by Theorem 2.4.19, and multiplying every matrix by \(\zeta \) multiplies every length-\(N\) word by \(\zeta ^{N}\) by Lemma 2.4.12, hence every length-\(N\) coefficient of \(B\) is \(\zeta ^{N}\) times that of \(A\). By Theorem 12.8.5 together with gauge invariance of the local ground space, \(G_L(A)=G_L(\zeta A)=G_L(B)\). The parent interaction then agrees by Theorem 12.8.6 and the periodic chain ground space by Lemma 11.6.5.
Let \(D{\gt}0\) and let \(A\) be a normal tensor. There exist \(l{\gt}1\), \(\epsilon _l\geq 0\), an integer \(m\geq l+1\), and \(\delta {\gt}0\) such that \(A\) is \(l\)-block injective, \(\epsilon _l{\lt}1/\sqrt{l+1}\),
and, for every \(N\geq 2m\) and every \(v\in (\ker H_{N,l+1}^{\mathrm{ES}}(A))^\perp \),
No normalization is imposed on \(A\). The uniform gap of matrix product state parent Hamiltonians is asserted at
[
17
, lines 2183–2187
]
; the arbitrary finite-range coefficient (359) is the derivation recorded in docs/paper-gaps/knabe88_finite_range_coefficient.tex.
Replace \(A\) by the normalized primitive representative \(B\) of Theorem 12.8.8 and apply Theorem 12.3.7.82 to \(B\). Since \(G_L(A)=G_L(B)\) for every \(L\), Theorem 12.8.6 makes the two parent Hamiltonians equal, so both the kernel and the norm bound transfer verbatim. Block injectivity of \(B\) at length \(l\) passes to \(\zeta A\) by Lemma 2.5.6 applied to the inverse gauge, and from \(\zeta A\) to \(A\) by Theorem 12.8.1.
Let \(P\) and \(Q\) be finite-dimensional complex endomorphisms, and let \(\gamma \geq 0\). Suppose that \(P\) is positive and
for every \(v\in (\ker P)^\perp \). Then
on \((\ker P)^\perp \). If also \(P\leq Q\) and \(\ker P=\ker Q\), then
for every \(v\in (\ker Q)^\perp \).
Let \((P_i)_{i\in I}\) be a finite family of positive operators on an inner product space. A positive operator \(P\) annihilates a vector \(v\) as soon as \(\operatorname {Re}\langle Pv,v\rangle =0\), and consequently, for real weights \(w_i{\gt}0\),
equivalently \((\sum _{i\in I}w_iP_i)v=0\) exactly when \(P_iv=0\) for every \(i\in I\). For the empty family both sums are zero and all three kernels are the full space. This is the analytic kernel step of the blocked-sum comparison in Lemma block-obc of Rozmán–Molnár–Schuch 1 , in which the blocked parent Hamiltonian is a weighted sum \(\sum _i\gamma _ih_i\) of positive local terms with \(\gamma _i{\gt}0\).
If \((\sum _iw_iP_i)v=0\), the real part of the inner product with \(v\) expands as
Each summand is nonnegative, so each vanishes, and \(w_i{\gt}0\) forces \(\operatorname {Re}\langle P_iv,v\rangle =0\). For a positive operator \(P\), the quadratic form is nonnegative along the real affine line \(t\mapsto v+t\cdot (Pv)\) and vanishes at \(t=0\); its linear coefficient is \(2\lVert Pv\rVert ^2\), so evaluating at small negative \(t\) forces \(\lVert Pv\rVert =0\), that is \(Pv=0\). The converse inclusion holds term by term, and the unweighted comparison is the case \(w_i=1\).
Let \((P_i)_{i\in I}\) be a finite family of positive operators on an inner product space, with real weights \((w_i)_{i\in I}\). If \(w_i\geq 0\) for every \(i\), then \(\sum _{i\in I}w_iP_i\) is positive. If \(w_i\leq 1\) for every \(i\), then
in Loewner order. This is the comparison step of the blocked-sum argument in the source cited in Theorem 12.8.11, where the coefficients satisfy \(0{\lt}\gamma _i\leq 1\).
Positivity of the weighted sum follows from positivity of each scalar multiple \(w_iP_i\) and of finite sums of positive operators. For the domination, the difference is again a weighted sum,
now with nonnegative weights \(1-w_i\geq 0\), hence positive.
Let \((P_i)_{i\in I}\) be a finite family of positive operators on a finite-dimensional complex inner product space, with real weights \(0{\lt}w_i\leq 1\), and let \(\gamma \geq 0\). If
for every \(v\in \bigl(\ker \sum _{i\in I}w_iP_i\bigr)^\perp \), then
for every \(v\in \bigl(\ker \sum _{i\in I}P_i\bigr)^\perp \).
For any tensor \(A\), interaction range \(L\), and chain length \(N\), there exists \(\delta _N{\gt}0\) such that \(\delta _N\lVert v\rVert \leq \lVert H_{N,L}^{\mathrm{ES}}(A)v\rVert \) for every \(v\in (\ker H_{N,L}^{\mathrm{ES}}(A))^\perp \).
The restriction of \(H_{N,L}^{\mathrm{ES}}(A)\) to its kernel complement is injective. Its inverse on its image is bounded, since the domain is finite dimensional. Thus \(\lVert v\rVert \leq C_N\lVert H_{N,L}^{\mathrm{ES}}(A)v\rVert \) for some \(C_N{\gt}0\); take \(\delta _N=C_N^{-1}\). If the kernel complement is zero, any positive constant suffices.
Fix \(A\) and \(L\). Suppose \(\delta {\gt}0\) and \(M\in \mathbb N\) satisfy \(\delta \lVert v\rVert \leq \lVert H_{N,L}^{\mathrm{ES}}(A)v\rVert \) for every \(N\geq M\) and every \(v\in (\ker H_{N,L}^{\mathrm{ES}}(A))^\perp \). Then some \(\gamma {\gt}0\) satisfies the inequality with \(\gamma \) in place of \(\delta \) for every \(N\).
Take the minimum of \(\delta \) and the finitely many positive constants \(\delta _N\) supplied by Theorem 12.8.14 for \(N{\lt}M\). Equivalently, induct on \(M\), replacing \(\delta \) by \(\min \{ \delta ,\delta _{M-1}\} \) at each step.
Let \(D{\gt}0\) and let \(A\) be a normal tensor. There exist \(l{\gt}1\) and \(\gamma {\gt}0\) such that \(A\) is \(l\)-block injective and, for every \(N\) and every \(v\in (\ker H_{N,l+1}^{\mathrm{ES}}(A))^\perp \),
This proves the normal-tensor case of the uniform gap assertion in [ 17 , lines 2183–2187 ] , at a suitable interaction range.
Let \(L{\gt}1\). Suppose that for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,
in quadratic form. Then the parent Hamiltonian has the explicit gap constant \(\gamma _L:=1/(4L){\gt}0\). Concretely, if \(N\ge 2L\) and \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), then \(\gamma _L\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).
The finite-overlap martingale reduction applies with the cyclic overlap relation: at most \(2(L-1)\) off-diagonal neighbours per row; for non-overlapping pairs the disjoint-window estimate \(\operatorname{Re}\langle h_{i,\mathrm{ES}}v,h_{j,\mathrm{ES}}v\rangle \ge 0\) holds; and the anticommutator inequality is retained as an explicit hypothesis.
For a tensor \(A\) and a range \(L{\gt}1\), assume the cyclic-window anticommutator estimate of Theorem 12.8.17. Then there exists \(\gamma {\gt}0\) such that for every \(N\ge 2L\) and every vector \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \) one has \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). By Lemma 12.3.20, this implies the usual uniform lower bound on every nonzero eigenvalue of the parent Hamiltonian.
This follows immediately from the corresponding explicit theorem by taking \(\gamma \) to be the constant produced there.
Fix \(p\geq 1\), \(M\geq 2\), and an open chain of length \(N=pM\). For \(0\leq n\leq M\), let \(G_n\) be the ground projection of the range-\(2p\) parent Hamiltonian on \([0,pn)\), extended by the identity on the remaining sites. For \(0\leq n{\lt}M\), let \(Q_n\) be the ground projection of the same interaction restricted to \([\max \{ 0,p(n-1)\} ,p(n+1))\). Thus \(Q_0=I\), while for \(n\geq 1\) the interval Hamiltonian is a single parent interaction.
Set \(E_n=G_n-G_{n+1}\). For \(0\leq m,n{\lt}M\) with \(m{\lt}n-1\) or \(m{\gt}n\), one has
The initial differences satisfy \(E_0=0\) and \(Q_1E_1=0\).
In the first case both prefix Hamiltonians have support disjoint from the interval defining \(Q_n\). In the second case their kernels are contained in the local interval kernel. The corresponding orthogonal projections therefore commute in either case. The prefixes of lengths zero and \(p\) contain no interaction, so \(G_0=G_1=I\). At the next step \(Q_1=G_2\), giving \(Q_1E_1=G_2(I-G_2)=0\).
Let \(p\geq 1\), let \(N\geq 2p\), and let \(M\geq 0\). Define \(G_n\), \(Q_n\), and \(E_n=G_n-G_{n+1}\) on the \(N\)-site chain by the formulas of Definition 12.8.19. Let \(\epsilon \geq 0\) and suppose that \(\| Q_nE_n\| \leq \epsilon \) for \(2\leq n{\lt}M\). Then \(\| Q_nE_n\| \leq \epsilon \) for \(0\leq n{\lt}M\).
By Lemma 12.8.20,
so both norms are \(0\leq \epsilon \). For \(2\leq n{\lt}M\) the bound is the hypothesis.
Let \(p\geq 1\), \(M\geq 2\), and \(N=pM\). Consider the range-\(2p\) open parent Hamiltonian \(H_N\) and the grouped projections above. Let \(0\leq \epsilon {\lt}1/\sqrt2\) and suppose that \(\| Q_nE_n\| \leq \epsilon \) for \(0\leq n{\lt}M\). Then, for every \(v\perp \ker H_N\),
Apply the full-range martingale estimate to the grouped filtration, with window width one. Each local window Hamiltonian is the projection \(I-Q_n\), so Theorem 12.8.24 gives condition C2 with constant one. The nonzero local window terms are distinct summands of \(H_N\), so Theorem 12.8.23 gives condition C1 with constant one. Lemma 12.3.7.4 supplies positivity of \(H_N\). The locality lemma supplies the commutation condition. The assumed norm bound supplies C3, and the terminal projection is the projection onto \(\ker H_N\).
Let \(p\geq 1\), \(M\geq 0\), and \(N=pM\). For every vector \(v\) on the \(N\)-site chain,
The indices \(p(n+1)-1\), for \(0\leq n{\lt}M\), are distinct and belong to \(\{ 0,\ldots ,pM-1\} \). Positivity bounds the sum over these indices by the full sum, which is bounded by the open-chain energy.
Let \(p\geq 1\), \(0\leq n{\lt}M\), and \(N=pM\). Write \(Q_n\) for the orthogonal projection onto the kernel of \(H^{\mathrm{suffix}}_{N,2p;2p,p(n+1)}(A)\). Then
for every vector \(v\) on the \(N\)-site chain.
Apply the full-range local gap bound with \(l=2p-1\) and index \(p(n+1)-1{\lt}N\).
Let \(R{\gt}0\) and suppose that the length-\(n\) open parent Hamiltonian has kernel \(G_n(A)\). On \(n+r\) sites, the kernel of the same interactions restricted to the first \(n\) sites is \(G_n(A)\otimes (\mathbb {C}^d)^{\otimes r}\). A vector belongs to this space if and only if each restriction obtained by fixing the last \(r\) physical coordinates belongs to \(G_n(A)\).
Decompose a vector \(v\) on \(n+r\) sites according to the last \(r\) physical coordinates, writing \(v_s\) for the vector on \(n\) sites obtained by fixing these coordinates to \(s\in \{ 0,\ldots ,d-1\} ^r\). The prefix Hamiltonian acts on each component \(v_s\) by the length-\(n\) open Hamiltonian, whose kernel is \(G_n(A)\). Hence \(v\in G_n(A)\otimes (\mathbb {C}^d)^{\otimes r}\) if and only if, for every \(s\in \{ 0,\ldots ,d-1\} ^r\),
Reassociation identifies any subspace of the physical space on \(K+(L+Q)\) sites with the same subspace on \((K+L)+Q\) sites. In particular, when \(L+Q{\gt}0\), the kernel of the single parent interaction on the last \(L+Q\) sites is \((\mathbb {C}^d)^{\otimes K}\otimes G_{L+Q}(A)\) in these common coordinates.
Write \(\sigma \) for the reassociation isometry from the physical space on \(K+(L+Q)\) sites to that on \((K+L)+Q\) sites. It preserves the ordered physical coordinates, so for every family \((S_n)_n\) of subspaces of the \(n\)-site physical spaces,
A parent interaction is the orthogonal projection onto the complement of the local boundary-condition space, acting trivially on the other sites. Its kernel on the last \(L+Q\) sites is therefore \((\mathbb {C}^d)^{\otimes K}\otimes G_{L+Q}(A)\), and the display transports this description to the common coordinates.
Let \(0{\lt}R\leq W\leq n\leq N\), and suppose that the range-\(R\) open parent Hamiltonian on \(W\) sites has kernel \(G_W(A)\). On the \(N\)-site chain, the range-\(R\) interaction sum on \([n-W,n)\) and the single parent interaction on the same interval have the same kernel. In particular, for \(W=L+Q\) and a chain of length \(K+L+Q\), the kernel of the terminal range-\(R\) interaction sum is \((\mathbb {C}^d)^{\otimes K}\otimes G_{L+Q}(A)\), the boundary space of condition (C3\('\)) in [ 225 ] .
The physical coordinate isometry identifies both interval operators with the corresponding \(W\)-site operators tensored with the identity on the remaining sites. Tensoring with the identity preserves equality of kernels, and the \(W\)-site kernels are both \(G_W(A)\). The terminal case then follows from the reassociated description of the kernel of the single parent interaction on the last \(L+Q\) sites.
Let \(R=L+Q{\gt}0\) and suppose that the range-\(R\) open parent Hamiltonians on \(K+L\) and \(K+L+Q\) sites have their boundary-condition spaces as kernels. On \(K+L+Q\) sites, let \(P\) project onto the ground space of the first \(K+L\) sites, let \(T\) project onto the ground space of the last \(L+Q\) sites, and let \(F\) project onto the whole-chain ground space. Then
The right-hand side is the three-interval projector error in [ 225 , condition C3\('\) ] .
The whole-chain boundary space lies in the suffix boundary space, so \(TF=F\). Thus \(T(P-F)=TP-F\). Taking adjoints, using that orthogonal projections are self-adjoint, and reversing the sign gives the identity.
Let \(A\) be a tensor, let \(p,R\geq 1\) be integers, and let \(\gamma {\gt}0\). Suppose that, for every sufficiently large integer \(M\) and every \(u\perp \ker H^{\mathrm{open}}_{pM,R}(A)\), \(\gamma \lVert u\rVert \leq \lVert H^{\mathrm{open}}_{pM,R}(A)u\rVert \). Then there is \(\delta {\gt}0\) such that, for every sufficiently large \(N\) and every \(v\perp \ker H^{\mathrm{ES}}_{N,R}(A)\), \(\delta \lVert v\rVert \leq \lVert H^{\mathrm{ES}}_{N,R}(A)v\rVert \).
Choose \(n\) with \((R-1)^2{\lt}n\gamma \) and then choose \(M\geq n+2R\) for which the open-chain bound holds. Set \(m=pM-R+1\). Then \(m\geq \max \{ n,R\} \) and the active open volume is \(m+R-1=pM\). The finite-range Knabe comparison gives the positive constant \(\delta =(m\gamma -(R-1)^2)/(m-R+1)\) and the asserted bound for every \(N\geq 2m\).
Let the interaction range satisfy \(R{\gt}0\), let \(p\leq n\), and add \(r\geq 0\) sites to the right of an active interval of length \(n\). In each ambient volume \(s\in \{ n,n+r\} \), write \(P^{(s)}_a\) for the kernel projection of the prefix Hamiltonian ending at \(a\), and write \(Q^{(s)}\) for the kernel projection of the suffix Hamiltonian of length \(l\) ending at \(n\). Then
Let \(\Phi \) be the isometry \(\ell ^2([d]^{n+r})\to \ell ^2([d]^n\times [d]^r)\) splitting a configuration into its active and spectator coordinates, and write \(T^{\oplus [d]^r}\) for the fiberwise extension of an operator \(T\) on \(n\) sites. By Theorem 12.3.4.15, \(\Phi \) conjugates each of the three Hamiltonians to its fiberwise extension. Lemma 12.3.4.13 transfers this to their kernel projections: for \(a\in \{ p,n\} \),
Since fiberwise extension respects composition and subtraction (Lemma 12.3.4.12),
Conjugation by \(\Phi \) preserves the operator norm, and \(\lVert T^{\oplus [d]^r}\rVert \leq \lVert T\rVert \) by Lemma 12.3.4.12.
Let \(p\geq 1\). Suppose that the range-\(2p\) open parent Hamiltonian of a tensor has its boundary-condition space as kernel at every length at least \(2p\). For every \(K\geq 0\), let \(P_K\), \(T_K\), and \(F_K\) be the boundary-space projections on the first \(K+p\) sites, the last \(2p\) sites, and all \(K+2p\) sites, respectively, with free sites elsewhere. Suppose that \(\epsilon \geq 0\) and that, for every \(K\geq 0\),
Then on every chain of length \(pM\) with \(M\geq 2\), the grouped martingale projections satisfy
for every \(0\leq n{\lt}M\).
The products at indices zero and one vanish, so by Theorem 12.8.21 it suffices to treat \(n\geq 2\). For such \(n\), put \(K=p(n-1)\). The active interval has length \(p(n+1)\), and its preceding prefix has length \(pn\). The kernel hypothesis and the active projector identity give the desired bound on this interval. Adjoining the remaining \(pM-p(n+1)\) sites does not increase the operator norm.
Let \(A_0,\ldots ,A_{r-1}\) be normalized primitive tensors with common physical dimension \(d\geq 1\), positive bond dimensions, and positive-definite invariant matrices. Assume that no two distinct blocks are related by a virtual similarity and a phase. Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\neq 0\). For every \(R_*\in \mathbb N\), there are \(p\geq \max \{ 1,R_*\} \) and \(\gamma {\gt}0\) such that
for every \(M\geq 2\) and every \(v\perp \ker H^{\mathrm{open}}_{pM,2p}(B)\). This follows from the grouped martingale estimate of [ 225 , Theorem 2.1(ii) and Section 6 ] .
Choose \(p\) above \(R_*\) and the open-kernel threshold, and sufficiently large that the three-interval projector error is at most \(1/2\) for every pair of outer lengths at least \(p\). The grouped C3 estimate then holds with \(\epsilon =1/2{\lt}1/\sqrt2\). The grouped open-chain bound gives \(\gamma =(1-\sqrt2/2)^2{\gt}0\) at every length \(pM\), \(M\geq 2\).
Under the hypotheses of Theorem 12.8.32, for every \(R_*\in \mathbb N\) there are \(p\geq \max \{ 1,R_*\} \) and \(\delta {\gt}0\) such that
for every chain length \(N\) and every \(v\perp \ker H^{\mathrm{open}}_{N,2p}(B)\). For \(N{\lt}2p\) the Hamiltonian is zero and the assertion is vacuous. This is the open-chain form of [ 225 , Theorem 1.2 ] .
Choose \(p\) as in Theorem 12.8.32, above the open-kernel threshold and so large that the three-interval projector error is at most \(1/2\). Write \(N=pM+q\) with \(0\leq q{\lt}p\) and \(M\geq 3\). If \(q=0\) the grouped bound applies. Otherwise, the prefix of length \(pM\) carries the grouped gap \(\gamma \), and the terminal interval of length \(2p+q\) carries a gap \(\eta {\gt}0\), the minimum of the finitely many positive gaps at lengths below \(3p\). The two intervals overlap in \(2p\) sites, their kernels are the boundary spaces of the three-interval estimate, and combining the interval gaps gives the bound \(\min \{ \gamma ,\eta \} /16\) at every \(N\geq 3p\). The finitely many remaining lengths have positive gaps on their kernel complements, so a uniform \(\delta {\gt}0\) exists.
Let \(A_0,\ldots ,A_{r-1}\) be normalized primitive tensors with common physical dimension \(d\geq 1\), positive bond dimensions, and positive-definite invariant matrices. Assume that no two blocks are related by a virtual similarity and a phase. Let \(\mu _j\ne 0\) for every \(j\), and set \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\). For every prescribed lower bound \(R_*\), there exist \(R\geq \max \{ 1,R_*\} \), \(\gamma {\gt}0\), and \(N_0\) such that the periodic range-\(R\) parent Hamiltonian satisfies
for every \(N\geq N_0\) and every \(v\perp \ker H_{R,N}\). In particular, the block sum admits a parent interaction with a uniform positive periodic gap.
Let \(T\) be an operator on a finite-dimensional physical space and let \((e_s)_s\) be an orthonormal basis for an additional physical space. For \(x=\sum _s x_s\otimes e_s\) and \(y=\sum _s y_s\otimes e_s\),
Consequently \(T\ge 0\) implies \(T\otimes \mathbb {1}\ge 0\), and \(T\le U\) implies \(T\otimes \mathbb {1}\le U\otimes \mathbb {1}\). Scalar multiplication commutes with extension: \((cT)\otimes \mathbb {1}=c(T\otimes \mathbb {1})\). If \(V\) is a unitary identification of physical spaces, then \(VTV^{-1}\le VUV^{-1}\) if and only if \(T\le U\).
Expand the inner product in the product orthonormal basis and sum first over the additional coordinates. Positivity follows term by term; the order assertion follows by applying positivity to \(U-T\). The scalar identity follows on each coordinate. Unitary conjugation preserves and reflects positivity, hence also operator order.
Let \(A\) be an MPS tensor, and let \(R,W\) be interaction and interval lengths. Write \(K_{R,W}=H^{\mathrm{open}}_{W,R}(A)\) for the open parent Hamiltonian of interaction length \(R\) on \(W\) sites, and suppose that its kernel is \(\mathcal G_W(A)\). Writing \(h_W\) for the orthogonal projection onto \(\mathcal G_W(A)^\perp \), there are positive constants \(\kappa ,C\) such that
The operator \(K_{R,W}\) is a sum of orthogonal projections, hence positive. By hypothesis its kernel is \(\mathcal G_W(A)\), so the orthogonal projection onto \((\ker K_{R,W})^\perp \) is \(h_W\). Lemma 12.8.37 applied to \(K_{R,W}\) gives the two bounds.
Let \(H\) be a positive operator on a finite-dimensional complex Hilbert space. If \(P\) is the orthogonal projection onto \((\ker H)^\perp \), then \(aP\leq H\leq bP\) for some \(a,b{\gt}0\). More precisely, a norm gap \(\gamma \geq 0\) on \((\ker H)^\perp \) implies \(\gamma P\leq H\).
The restriction of \(H\) to \((\ker H)^\perp \) is injective and thus has a positive norm lower bound. Since \(H\) is positive, its eigenvalues on this subspace have the same lower bound. The operator norm gives the upper bound. Orthogonal projection onto \((\ker H)^\perp \) leaves the quadratic form of \(H\) unchanged.
If \(0\leq P\leq Q\), then \(\ker Q\subseteq \ker P\). Consequently, if \(P,Q\) are positive operators and \(a,b,c{\gt}0\) satisfy
their kernels agree. In finite dimension, a norm gap \(\gamma {\gt}0\) for \(P\) on its kernel complement implies a norm gap \(a\gamma /b\) for \(Q\) on the same subspace.
A positive quadratic form vanishes only on the kernel of its operator. Evaluating the order bounds on kernel vectors proves the inclusions. On the common kernel complement, the norm gap for \(P\) is equivalent to \(P\geq \gamma I\). Hence \(bQ\geq a\gamma I\), which gives the stated norm gap by Cauchy–Schwarz.
Let \(A\) be an MPS tensor, and let \(R,W\) be positive integers with \(W\geq 2R\). Suppose that the open range-\(R\) Hamiltonian on \(W\) sites has kernel \(\mathcal G_W(A)\). If the periodic parent Hamiltonians of range \(W\) have a uniform positive gap for all sufficiently large chain lengths, then the same holds at range \(R\).
By Lemma 12.8.36, choose \(\kappa ,C{\gt}0\) such that \(\kappa h_W\leq K_{R,W}\leq Ch_W\). Summing all cyclic translates (Theorem 12.8.41) gives
for sufficiently large \(N\). By Lemma 12.8.38, the periodic kernels therefore agree, and a gap \(\gamma \) at range \(W\) transfers to the gap \(\kappa \gamma /(W-R+1)\) at range \(R\).
For an MPS tensor \(A\) and an interval of positive length \(W\), the length-\(W\) parent interaction starting at the first site is the orthogonal projection \(h_W\) onto \(\mathcal G_W(A)^\perp \).
Restriction to the full interval is the identity, so the local interaction is precisely the defining orthogonal projection.
Let \(1\leq R\leq m\), \(N\geq 2m\), and \(W=m+R-1\). Write \(K_{R,W}\) for the open parent Hamiltonian of interaction length \(R\) on \(W\) sites, and \(H_{R,N}\) for the corresponding periodic Hamiltonian on \(N\) sites. If real constants \(\kappa ,C\) satisfy
where \(h_W\) projects onto \(\mathcal G_W(A)^\perp \), then
Extend each local inequality by the identity on the complementary sites and conjugate by the cyclic identification of the chosen interval. Both operations preserve operator order. Summing over all cyclic origins gives the asserted bounds: each range-\(R\) interaction occurs exactly \(m\) times, once for each of its possible positions in the open interval.
Let \(B=\bigoplus _j\mu _j A_j\), where the coefficients are nonzero and \((A_j)_j\) is a finite family of normalized primitive tensors of positive bond dimension, pairwise inequivalent under similarity and phase. Assume that each recorded invariant density is positive definite and that the physical dimension is positive. There exists \(R_0\geq 1\) such that, for every \(R\geq R_0\), there exist \(\gamma _R{\gt}0\) and \(N_R\) for which
for every \(N\geq N_R\) and every \(v\perp \ker H_{R,N}(B)\).
Choose \(R_0\) so that every open parent Hamiltonian of range at least \(R_0\) has the expected local ground space. Fix \(R\geq R_0\), and choose a uniformly gapped range \(W\geq 2R\). On \(W\) sites, the range-\(R\) Hamiltonian and the orthogonal projection onto the complement of the local MPS space have the same kernel. Finite dimensionality gives two-sided positive order bounds. Summing their cyclic translates gives two-sided bounds between the periodic Hamiltonians of ranges \(R\) and \(W\). These bounds identify their kernels and transfer the positive gap to range \(R\).
Let \(B=\bigoplus _{j=0}^{r-1}\mu _j A_j\), where \(r\geq 2\), all coefficients are nonzero, and the normalized primitive tensors \(A_j\) are pairwise inequivalent under similarity and phase. Assume positive physical and bond dimensions, positive definite recorded invariant densities, and a common block-injectivity length \(L_0\geq 1\). For every \(R\geq 3(r-1)(L_0+1)+1\), there exist \(\gamma _R{\gt}0\) and \(N_R\) such that
for every \(N\geq N_R\) and every \(v\perp \ker H_{R,N}(B)\). This removes the restriction to the range constructed by the overlap argument, as recorded in [ 131 ] .
The common block-injectivity hypothesis identifies the kernel of the open-chain range-\(R\) Hamiltonian on every interval of length at least \(R\) with the local MPS space. Choose a uniformly gapped range \(W\geq 2R\). The two-sided comparison on the interval of length \(W\) and its cyclic translates transfers the gap from range \(W\) to range \(R\).
Let \(B=\bigoplus _{j=0}^{r-1}\mu _j A_j\), where \(r\geq 2\), all coefficients are nonzero, and the normalized primitive tensors \(A_j\) are pairwise inequivalent under similarity and phase. Assume positive physical and bond dimensions, positive definite recorded invariant densities, and a common block-injectivity length \(L_0\geq 1\). For every \(R\geq 3(r-1)(L_0+1)+1\), there exists \(\gamma _R{\gt}0\) such that, for every \(N\in \mathbb N\) and every \(v\perp \ker H_{R,N}(B)\),
The eventual uniform gap leaves only finitely many chain lengths untreated. At each such length the Hamiltonian, restricted to its kernel complement, has a positive lower norm bound. Taking the minimum of these bounds and the eventual uniform constant proves the assertion.
Let \(A\) be primitive with nonzero bond dimension and fixed point \(\rho \). Write \(E=E_{\mathrm{FNW},A}\) for its observable transfer map and \(P=E_\infty ^\rho \) for its stationary projection. Every nonzero eigenvalue of \(E-P\) is an eigenvalue of \(E\).
The identities \(PE=P\) and \(P^2=P\) imply \(P(E-P)=0\). If \((E-P)X=\nu X\) with \(X\neq 0\) and \(\nu \neq 0\), applying \(P\) gives \(PX=0\), hence \(EX=\nu X\).
Let \(A\) be primitive with nonzero bond dimension and positive-definite fixed point \(\rho \). Suppose \(\lambda {\gt}0\) and every eigenvalue \(\nu \neq 1\) of its observable transfer map satisfies \(|\nu |{\lt}\lambda \). Then the spectral radius of the transfer remainder in the \(\rho \)-weighted norm is less than \(\lambda \). If \(\operatorname{tr}\rho =1\), there is a constant \(C_\lambda {\gt}0\) such that, for every integer \(n\geq 1\),
This establishes the eigenvalue prescription following [ 225 , Section 6, equation (6.1) ] .
A nonzero remainder eigenvalue is a transfer eigenvalue, and primitivity excludes the eigenvalue \(1\). The zero eigenvalue also has modulus less than \(\lambda \). In finite dimension these strict eigenvalue bounds give a strict spectral-radius bound, to which the prescribed-rate mixing estimate applies.
Let \(A\) be primitive with nonzero bond dimension and a positive-definite trace-one fixed point \(\rho \). Suppose \(\lambda {\gt}0\) exceeds the modulus of every nonunit transfer eigenvalue. For every \(c{\gt}0\), there is an integer \(M\) such that, for every \(n\geq M\),
The onset \(M\) may depend on the state, rate, and prescribed prefactor.
Choose an intermediate rate \(\mu \) strictly between the remainder spectral radius and \(\lambda \). The mixing estimate gives \(a(n)\leq C_\mu \mu ^n\). Since \((\mu /\lambda )^n\) tends to zero, eventually \(C_\mu (\mu /\lambda )^n\leq c\).
Under the preceding hypotheses, suppose additionally that \(\lambda {\lt}1\). For every \(c{\gt}0\), there exists an onset \(M\) such that, for every overlap \(m\geq M\), every prefix length \(r\geq 0\), and every suffix length \(\ell \geq 1\), we have \(c\lambda ^m{\lt}1\) and
The onset depends on the state, rate, and prefactor; it is not asserted to be the injectivity threshold. This is the version of [ 225 , Section 6, equation (6.1) ] with the additional onset stated explicitly [ 135 ] .
Increase the onset until the overlap exceeds an injectivity length and \(c\lambda ^m{\lt}1\). The preceding mixing estimate then satisfies the hypotheses of the fixed-overlap FNW projector estimate, uniformly in the two exterior lengths.
12.8.1 The dimension-only prefactor at a prescribed rate
Let \(t=(t_{ij})\) and \(z=(z_{ij})\) be real matrices indexed by a finite set of \(k\) vertices. Set \(A^{ij}=t_{ij}E_{ij}\). For endpoints \(a,f\), define the four-site vector
Write \(\mathbf1\) for the mark with all entries one and, for a real function \(u\) on the vertices, \(u\otimes \mathbf1\) and \(\mathbf1\otimes u\) for the marks \(z_{be}=u_b\) and \(z_{be}=u_e\). For \(X\in M_{k}(\mathbb {C})\), the four-site boundary vector of \(A\) has the coordinates
Writing \(T_{abcef}=t_{ab}t_{bc}t_{ce}t_{ef}\), one has
For every configuration \(\sigma =(i_0j_0,i_1j_1,i_2j_2,i_3j_3)\) and every real function \(u\) on the vertices,
Distinct vertex paths \((b,c,e)\) give distinct physical basis vectors \(|ab,bc,ce,ef\rangle \), which gives the first identity. For the second, multiply four consecutive letters:
Hence \(\Gamma _4(X)(ab,bc,ce,ef)=T_{abcef}X_{fa}\) on every vertex path. On the support of \(v_{a,f}\), the first letter \(ab\) determines \(b=j_0\) and the last letter \(ef\) determines \(e=i_3\), which gives the two mark identities.
Let \(K,L\in \mathbb {N}\) and \(X\in M_{D}(\mathbb {C})\). Let \(\varphi \) be a complex function of the prefix configurations \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) and \(\psi \) a complex function of the suffix configurations \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \). Then the vector \((u,\tau )\mapsto \varphi (u)\, \Gamma _{K+L}(X)(u,\tau )\) belongs to the range of \(\Gamma ^{\mathrm{tail}}_{K,L}\), and the vector \((u,\tau )\mapsto \psi (\tau )\, \Gamma _{K+L}(X)(u,\tau )\) belongs to the range of \(\Gamma ^{\mathrm{left}}_{K,L}\).
Factor the matrix product at the boundary of the interval:
where \(Y_u=\varphi (u)XA^u\) and \(Z_\tau =\psi (\tau )A^\tau X\).
For the weighted matrix-unit tensor \(A^{ij}=t_{ij}E_{ij}\) and all endpoints \(a,f\), one has \(v_{a,f}(\mathbf1)=\Gamma _4(E_{fa})\).
For every configuration,
The nonzero coordinates of \(\Gamma _4(E_{fa})\) are therefore those of the paths \((ab,bc,ce,ef)\), with value \(t_{ab}t_{bc}t_{ce}t_{ef}\). These are the coordinates of \(v_{a,f}(\mathbf1)\).
For every real function \(u\) on the vertices and all endpoints \(a,f\),
By Theorems 12.8.1.2 and 12.8.1.4, for \(\sigma =(i_0j_0,i_1j_1,i_2j_2,i_3j_3)\),
The first multiplier depends only on the first site and the second only on the last site. Theorem 12.8.1.3 with \((K,L)=(1,3)\) and \((K,L)=(3,1)\) gives the two memberships.
For a real weighted matrix-unit tensor, fix external vertices \(a,f\) and write \(p_{bce}=t_{ab}t_{bc}t_{ce}t_{ef}\). The path vector with mark \(z_{be}\) has norm one if \(\sum _{b,c,e}z_{be}^2p_{bce}^2=1\). It is orthogonal to the full four-site ground space if \(\sum _{b,c,e}z_{be}p_{bce}^2=0\).
Expand its self-inner product and its inner product with an arbitrary boundary vector. The latter equals the indicated scalar sum multiplied by the boundary matrix entry at \((f,a)\).
Let \(U,V,W\) be subspaces of a complex Hilbert space with orthogonal projections \(P_U,P_V,P_W\). If \(x\in V\cap W^\perp \) and \(y\in U\) have norm one, then
Since \(P_Vx=x\), \(P_Wx=0\) and \(P_Uy=y\), the inner product on the left equals \(\langle y,(P_UP_V-P_W)x\rangle \). Apply Cauchy–Schwarz and the operator norm bound. The two operator norms agree by taking adjoints.
For \(c=16\), \(\lambda =1/1000\), and \(m=2\), one has \(c\lambda ^m{\lt}1\) and
Here \(c\lambda ^m=1/62500\), and the displayed rational coefficient equals \(62501/(62500\cdot 62499)\).
Let \(e=(1,1,1,1)^{\mathsf T}\) and let \(u=(1,1,-1,-1)^{\mathsf T}\), \(v=(1,-1,1,-1)^{\mathsf T}\), \(w=(1,-1,-1,1)^{\mathsf T}\). Put \(Q=ee^{\mathsf T}/4\) and \(P=Q+(uv^{\mathsf T}+vw^{\mathsf T})/16\). Then \(P\) is strictly positive and doubly stochastic, and
This is an explicit construction for examining the prescribed-rate coefficient in [ 225 , Section 6, equation (6.1) ] .
The four displayed sign vectors are pairwise orthogonal and have squared norm four. Multiplication gives the identities. Each entry of \(P\) belongs to \(\{ 1/8,1/4,3/8\} \); its row and column sums are one.
Regard \(P\), \(u\) and \(w\) above as real. Then \(P_{ij}{\gt}0\), \(\sum _jP_{ij}=\sum _jP_{ji}=1\) and \(u_i^2=w_i^2=1\) for every \(i\), and the identities \(Pu=0\), \(w^{\mathsf T}P=0\), \(P^2=Q+uw^{\mathsf T}/64\) and \(P^3=Q\) of Theorem 12.8.1.9 hold over \(\mathbb {R}\). For every pair of endpoints \(a,f\), set \(p_{abcef}=P_{ab}P_{bc}P_{ce}P_{ef}\). Then
The amplitudes \(t_{ij}=\sqrt{P_{ij}}\) are strictly positive and satisfy \(t_{ij}^2=P_{ij}\).
The real identities are those of Theorem 12.8.1.9, read in \(\mathbb {R}\). For the path sums, apply the identities for \(P^2\) and \(P^3\), and use \(u_b^2=w_e^2=1\). The assertions concerning square roots follow from positivity.
For the transition matrix \(P\) and weighted matrix-unit tensor \(A^{ij}=\sqrt{P_{ij}}E_{ij}\) defined above, let \(G_I\) denote the orthogonal projection onto the ground space on the interval \(I\), extended by the identity on the remaining sites. Then
Fix external vertices \(a,f\) and write \(T_{abcef}=\sqrt{P_{ab}P_{bc}P_{ce}P_{ef}}\). Consider
With \(t_{ij}=\sqrt{P_{ij}}\) these are \(x=2v_{a,f}(u\otimes \mathbf1)\) and \(y=2v_{a,f}(\mathbf1\otimes w)\). By Theorem 12.8.1.5, \(x\) belongs to the range of \(G_{[1,3]}\) and \(y\) belongs to the range of \(G_{[0,2]}\). The path contraction identities give \(\| x\| =\| y\| =1\), \(x\perp \operatorname {range}G_{[0,3]}\), and \(\langle y,x\rangle =1/16\). Thus the matrix coefficient of \(G_{[0,2]}G_{[1,3]}-G_{[0,3]}\) between \(y\) and \(x\) is \(1/16\). Taking the adjoint gives the asserted order of the two projections.
Let \(A_{ij}=t_{ij}E_{ij}\), where \(t\) is a real matrix. If every entry of \(t\) is nonzero, then the letters of \(A\) span the full matrix algebra. If the squared entries in every column sum to one, then
When \(t_{ij}\ne 0\), the matrix unit \(E_{ij}\) is a scalar multiple of \(A_{ij}\). The normalization follows from \(A_{ij}^*A_{ij}=t_{ij}^2E_{jj}\) and the column-sum assumption.
Let \(t=(t_{ij})\) be a real matrix and define physical letters \(A_{ij}=t_{ij}E_{ij}\). The associated transfer map satisfies
Multiply the matrix units in \(\sum _{i,j}A_{ij}XA_{ij}^{\dagger }\). Only the diagonal entries of \(X\) contribute.
For the four-dimensional matrix \(P\) above, let \(A_{ij}=\sqrt{P_{ij}}E_{ij}\). Then \(A\) is injective at length one, its transfer map \(\mathcal E_A\) is unital and trace preserving, and every matrix \(X\) satisfies
These are the channel properties of the explicit construction used to examine [ 225 , Section 6, equation (6.1) ] .
The positive coefficients ensure that the physical letters span all matrix units. Column stochasticity gives trace preservation, and row stochasticity gives unitality. On diagonal matrices the transfer map acts by \(P\), while it annihilates off-diagonal matrix units. Therefore \(P^3=ee^{\mathsf T}/4\) gives the asserted formula.
For the weighted matrix-unit tensor of the Markov example, the matrix \(\rho =I/4\) is positive definite, has trace one, and is stationary. The tensor has a complementary transfer spectral gap at \(\rho \). Every nonunit eigenvalue of its FNW observable transfer map is zero.
The row sums of the Markov matrix give stationarity of \(I/4\). For a nonunit eigenvalue \(\nu \) of the trace-preserving transfer map, trace preservation forces an associated eigenvector \(X\) to have trace zero. The depolarizing third iterate then gives \(\nu ^3X=0\), and hence \(\nu =0\). The bilinear trace adjoint has the same eigenvalues, giving the assertion for the FNW observable transfer map. One-site injectivity gives irreducibility. Together with the unique peripheral eigenvalue and the faithful stationary density, this implies the complementary spectral gap.
There exists an injective matrix-product tensor of physical dimension \(16\) and auxiliary dimension \(4\) with a faithful primitive stationary density matrix of trace one such that every eigenvalue other than \(1\) of its observable transfer map has modulus smaller than \(\lambda =1/1000\). Nevertheless, at overlap length \(m=2\),
Thus the dimension-only coefficient and the prescribed-rate assertion following Nachtergaele’s bound cannot hold simultaneously at this fixed overlap length; see [ 135 ] .
Take the weighted matrix-unit tensor associated with the transition matrix above and the stationary density \(I/4\). Its transfer map becomes completely depolarizing after three steps, so every nonunit eigenvalue of the observable transfer map is zero. The rate \(1/1000\) is therefore admissible. The physical projection defect is at least \(1/16\), whereas the proposed coefficient is strictly smaller than \(1/16\).
Let \(A^0,\ldots ,A^{r-1}\) be tensors with bond dimensions \(D_0,\ldots ,D_{r-1}\), and let \(L\in \mathbb N\). Suppose that their simultaneous word evaluations at length \(L\) span \(\prod _{k=0}^{r-1}M_{D_k}(\mathbb {C})\). Then every \(A^j\) is \(L\)-block injective.
Let \(M\in M_{D_j}(\mathbb {C})\) and let \(X\) be the tuple with \(X_j=M\) and \(X_k=0\) for \(k\neq j\). By hypothesis,
for some coefficients \(c_\omega \). Its \(j\)-th component is \(M=\sum _{|\omega |=L}c_\omega A^{j,\omega }\), so the length-\(L\) words of \(A^j\) span \(M_{D_j}(\mathbb {C})\).
Let \(A^0,\ldots ,A^{r-1}\) be tensors with positive bond dimensions. Suppose that their simultaneous word evaluations at length \(S\) span the product of their matrix algebras. Then no two distinct blocks are gauge-phase equivalent. This is a consequence of the simultaneous block-injectivity condition in [ 17 , Section IV.C ] .
The local ground spaces of the blocks at length \(S\) are independent. Each block is \(S\)-block injective by Theorem 12.8.1.17, so each boundary map at that length is injective, and its value at the identity matrix, the periodic matrix product vector \(V^{(S)}(A^i)=\Gamma ^{A^i}_S(\mathbb {1})\), is nonzero. Suppose that \(i\neq j\) and \(A^{i,a}=\zeta XA^{j,a}X^{-1}\) for every physical index \(a\). Then
so \(\Gamma ^{A^i}_S(\mathbb {1})=0\), contradicting injectivity of the boundary map.
Let \(A^0,\ldots ,A^{r-1}\) have positive bond dimensions and a common physical dimension. Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Then their word tuples span the same algebra at every length \(n\geq S\). This is the propagation of block injectivity used in [ 17 , Section IV.C ] .
Each block is \(S\)-block injective and hence normal. Rescale and conjugate each block into normalized primitive form. Both operations preserve the simultaneous span at \(S\). Trace preservation propagates that span to \(n\geq S\). Inverse conjugation and rescaling return the full span to the original family at length \(n\).
Let \(A^0,\ldots ,A^{r-1}\) have positive physical and bond dimensions. Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Set \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\neq 0\). For every \(S+1\leq R\leq N\), \(\ker H^{\mathrm{open}}_{N,R}(B)=G_N^{\mathrm{ES}}(B)\). This sufficient range follows from the inversion and regrowth argument in [ 17 , Section IV.C, lines 2114–2129 ] .
Normalize the blocks while preserving their simultaneous span at \(S\) and every component local space. Trace preservation gives the span at all lengths at least \(S\), so the open-chain intersection theorem gives the kernel identity for the normalized block sum. The original and normalized sums have equal local spaces at \(R\) and \(N\); hence their open Hamiltonians and their spaces \(G_N^{\mathrm{ES}}\) agree.
Let \(A^0,\ldots ,A^{r-1}\) have positive bond dimensions and common physical dimension \(d\geq 1\). Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Let \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\ne 0\). For every interaction range \(R\geq S+1\), there exists \(\delta {\gt}0\) such that
for every \(N\geq R\) and every \(v\perp G_N(B)\). No normalization or inequivalence assumption on the original blocks is required. This is the finite-interval conclusion of [ 225 , Theorem 1.2 and Section 6 ] at the sufficient interaction ranges supplied by block injectivity.
Independently normalize the blocks into primitive form while preserving their simultaneous span at \(S\) and all component local ground spaces. Simultaneous injectivity separates these normalized blocks. The all-length primitive theorem gives a uniform gap at some range \(2p\) with \(p\geq R\). At range \(R\), the open kernel is \(G_N\) for every \(N\geq R\). On the interval of length \(2p\), finite-dimensional comparison gives \(\kappa h_{2p}\leq H^{\mathrm{open}}_{2p,R}\) for some \(\kappa {\gt}0\). The interval comparison transfers the long-range gap to range \(R\) at every \(N\geq 2p\), with loss \(2p-R+1\). A finite minimum includes \(R\leq N{\lt}2p\). Equality of the local spaces returns the same Hamiltonians and ground spaces for the original weighted block tensor.
Let \(A^j\) be normal tensors with common physical dimension \(d\geq 1\), pairwise inequivalent under virtual similarity and phase. Set \(D_* = \max \{ \sum _j D_j,1\} \) and \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\ne 0\). For every \(R\geq 3D_*^5\), there exists \(\delta {\gt}0\) such that \(\delta \| v\| \leq \| H^{\mathrm{open}}_{N,R}(B)v\| \) for every \(N\geq R\) and every \(v\perp G_N(B)\).
The dimension-bound blocking theorem gives a positive simultaneous injectivity length \(S\) with \(S+1\leq 3D_*^5\). Apply the open-chain gap theorem at \(R\geq S+1\).
Let \(A\) be a canonical tensor with physical dimension \(d\geq 1\), and choose its distinct normal representatives \(B^j\). At every positive interaction range, its open parent Hamiltonian agrees with that of \(\bigoplus _j B^j\), including all multiplicities and the ambient reconstruction of \(A\). If the representatives are simultaneously injective at \(S{\gt}0\) and \(R\geq S+1\), then \(\ker H^{\mathrm{open}}_{N,R}(A)=G_N(A)\) for every \(N\geq R\), and there is \(\delta {\gt}0\) such that
for every \(N\geq R\) and every \(v\perp G_N(A)\). If the bond dimension \(D\) is positive, the conclusion holds at every \(R\geq 3D^5\) without a supplied injectivity length.
The canonical reconstruction preserves every positive-length local ground space. This identifies the open parent Hamiltonians and their boundary-condition spaces with those of the distinct representatives. Apply the block open-chain gap and kernel theorems. For the dimension bound, the blocking estimate supplies \(S{\gt}0\) with \(S+1\leq 3D^5\).
For a local operator \(h\) on \(R\) sites, let \(H^{\mathrm{open}}_N(h)\) be the sum of its translates over starts \(i\) satisfying \(0\leq i\) and \(i+R\leq N\). Each translate acts as the identity on the remaining physical sites.
Open interaction sums preserve order and positivity, commute with scalar multiplication, and vanish for the zero interaction. When \(R{\gt}0\), the sum of the canonical parent projection is the canonical open parent Hamiltonian.
Extension by the identity and physical coordinate isometries preserve order, positivity, and scalar multiplication. Summation preserves these properties. Extending the local parent projection gives each canonical parent term.
Let \(R{\gt}0\), and let \(h\geq 0\) have kernel \(G_R(A)\). There are \(\kappa ,C{\gt}0\), independent of \(N\), such that
Hence a uniform canonical open-chain gap \(\gamma {\gt}0\) for \(N\geq R\) gives the gap \(\kappa \gamma \) for \(H^{\mathrm{open}}_N(h)\) on its kernel complement. In particular, this applies to every fixed positive parent interaction of a canonical tensor when its normal representatives are simultaneously injective at \(S{\gt}0\) and \(R\geq S+1\).
Finite-dimensional positivity and the prescribed local kernel give \(\kappa h_R(A)\leq h\leq C h_R(A)\). Sum the translated comparisons. The two positive operators bound one another by positive constants, so their kernels coincide. The lower bound transfers the canonical gap to the common kernel complement.
Let \(K\) be a compact parameter set. Suppose the local ground projections of \(A_x\) are continuous, and let \(h_x\) be a continuous family of positive interactions with kernel \(G_R(A_x)\), where \(R{\gt}0\). If the canonical open parent Hamiltonians have one positive gap for every \(x\in K\) and every \(N\geq N_0\), then the interaction sums \(H^{\mathrm{open}}_N(h_x)\) also have one positive gap for those parameters and lengths.
For a fixed canonical tensor \(A\), this conclusion requires no supplied canonical gap: simultaneous injectivity at \(S{\gt}0\) and \(R\geq S+1\) give one gap uniform over \(x\in K\) and every \(N\geq R\). When its bond dimension \(D\) is positive, \(R\geq 3D^5\) suffices without a supplied injectivity length.
Compactness and continuity give a single \(\kappa {\gt}0\) with \(\kappa h_R(A_x)\leq h_x\) for every \(x\in K\). Each interaction also has a finite positive upper comparison constant. The preceding proof transfers the supplied gap with the common lower constant; the upper constants identify the kernels and do not affect the resulting gap. For a fixed canonical tensor, apply its all-length open gap theorem.
Let \(K\) be a compact parameter set, and let \(A_x^j\) be continuous block tensors with positive physical and bond dimensions. Suppose that their length-\(S\) word tuples span the product matrix algebra for every parameter, where \(S{\gt}0\). Set \(B_x=\bigoplus _j\mu _j(x)A_x^j\), with all weights nonzero; the weights need not be continuous. Fix \(R\geq S+1\) and a finite upper length \(M\). There is \(\delta {\gt}0\) such that
for every \(x\in K\), every \(R\leq N\leq M\), and every \(v\perp \ker H^{\mathrm{open}}_{N,R}(B_x)\). The constant may depend on \(M\).
At each fixed \(N\geq R\), simultaneous injectivity gives continuous local ground projections at lengths \(R\) and \(N\). The open Hamiltonian depends continuously on the first projection, and its kernel is \(G_N\). The compact finite-dimensional norm-gap theorem supplies one positive bound at this \(N\). Taking the minimum over \(R\leq N\leq M\) gives the claim. Nonzero weights preserve every local ground space.
Let \(R,P{\gt}0\). On the Hilbert space of an \(N\)-site chain, index the nonwrapping canonical interactions by \(s\in \mathbb {Z}/P\mathbb {Z}\). Writing \(\bar s\) for the representative in \(\{ 0,\ldots ,P-1\} \), define
The period indexes the interactions; the physical chain still has \(N\) sites.
Every \(\widetilde h_s\) is an orthogonal projection. If \(R\leq e\) and \(e+R\leq P\), then \(\widetilde h_s\widetilde h_{s+e} =\widetilde h_{s+e}\widetilde h_s\) for every \(s\). If \(N\leq P\), \(\sum _s\widetilde h_s=H^{\mathrm{open}}_{N,R}(A)\).
Each term is a local projection or zero. At the stated separation, either \(\bar s+R\leq \overline{s+e}\) or \(\overline{s+e}+R\leq \bar s\). When both terms are active, their nonwrapping intervals are disjoint, so the local terms commute. For \(P\geq N\), every nonwrapping start occurs exactly once; all remaining summands vanish.
Let \(m{\gt}0\), \(N\geq R{\gt}0\), and \(P=N+2m\). Put \(A_s=\sum _{q=0}^{m-1}\widetilde h_{s+q}\). Each \(A_s\) is zero or the range-\(R\) open Hamiltonian on an interval \([n-W,n)\), extended by the identity on the remaining sites, with \(R\leq W\leq m+R-1\) and \(W\leq n\leq N\). If every open Hamiltonian on \(R\leq W\leq m+R-1\) sites has norm gap at least \(\gamma {\gt}0\) on its kernel complement, then \(A_s^2\geq \gamma A_s\) for every \(s\).
The active starts are \(0,\ldots ,N-R\). Padding by \(2m\) zeros prevents a cyclic window of \(m\) starts from meeting two separated parts of this interval. Its active starts therefore form one interval \([a,b)\), or none. In the first case, put \(W=b-a+R-1\) and \(n=b+R-1\). The local terms sum to the interval Hamiltonian on \([a,n)\). Its norm gap is preserved under extension by free sites. Positivity converts this norm gap into \(A_s^2\geq \gamma A_s\); the zero case is immediate.
Let \(1\leq R\leq m\), and suppose that every range-\(R\) open parent Hamiltonian on \(R\leq W\leq m+R-1\) sites has norm gap at least \(\gamma \) on its kernel complement. If \(m\gamma {\gt}(R-1)^2\), then
and \(\delta \| v\| \leq \| H^{\mathrm{open}}_{N,R}(A)v\| \) for every \(N\geq R\) and \(v\perp \ker H^{\mathrm{open}}_{N,R}(A)\). This is a consequence of the cyclic finite-range inequality, as derived in [ 56 ] ; it is not the printed periodic nearest-neighbor criterion.
For each physical volume \(N\), take period \(P=N+2m\). The padded projections have sum \(H^{\mathrm{open}}_{N,R}(A)\) and satisfy the separated commutation hypothesis. Every cyclic window satisfies \(A_s^2\geq \gamma A_s\). The cyclic inequality therefore gives \((H^{\mathrm{open}}_{N,R}(A))^2\geq \delta H^{\mathrm{open}}_{N,R}(A)\) with \(\delta \) as in (396). The positive-operator quadratic-form estimate gives the asserted norm gap.
Let \(A_x^j\) be continuous block tensors with positive physical and bond dimensions. Suppose their length-\(S\) word tuples span the product matrix algebra at every parameter, where \(S{\gt}0\), and put \(B_x=\bigoplus _j\mu _j(x)A_x^j\), with every weight nonzero. Fix \(R\geq S+1\) and a finite upper volume \(M\). If at \(x_0\) the open Hamiltonians on every \(R\leq N\leq M\) have norm gap at least \(\gamma {\gt}0\) on \(G_N(B_{x_0})^\perp \), then every \(\delta {\lt}\gamma \) is a common lower bound on \(G_N(B_x)^\perp \) for those volumes and all \(x\) in some neighborhood of \(x_0\). The same assertion holds for one fixed \(N\geq R\). No continuity of the weights is required. This is the finite-volume continuity step of [ 236 , Appendix A, lines 2575–2578 ] .
The ground projections at \(R\) and \(N\) are continuous. The open Hamiltonian is a finite sum of translates of \(I-P_{G_R}\), hence continuous. Its ground space contains \(G_N\). The operator and projection perturbation estimate preserves each strict lower bound at fixed \(N\). Intersect the finitely many resulting neighborhoods.
Under the block-family hypotheses of Theorem 12.8.1.33, suppose that at \(x_0\) one \(\gamma {\gt}0\) bounds the open norm gaps on \(G_N(B_{x_0})^\perp \) for every \(N\geq R\). Then some \(\delta {\gt}0\) bounds the open norm gaps on their kernel complements for every \(N\geq R\) and every \(x\) in one neighborhood of \(x_0\).
Choose \(m\geq R\) with \(m\gamma /2{\gt}(R-1)^2\). Continuity preserves the bound \(\gamma /2\) on every \(R\leq W\leq m+R-1\) in one neighborhood. Simultaneous injectivity identifies each kernel with \(G_W(B_x)\). The finite-size criterion gives the same positive \(\delta =(m\gamma /2-(R-1)^2)/(m-R+1)\) for all \(N\geq R\) there.
Let \(K\) be compact, and let \(A_x^j\) be continuous block tensors with positive physical and bond dimensions. Suppose their length-\(S\) word tuples span the product matrix algebra for every \(x\), with \(S{\gt}0\). Put \(B_x=\bigoplus _j\mu _j(x)A_x^j\), with every weight nonzero. For each fixed \(R\geq S+1\), there is \(\delta {\gt}0\) such that \(\delta \| v\| \leq \| H^{\mathrm{open}}_{N,R}(B_x)v\| \) for every \(x\in K\), every \(N\geq R\), and every \(v\perp \ker H^{\mathrm{open}}_{N,R}(B_x)\). No continuity of the weights is required. The compactness argument follows [ 236 , Appendix A ] at the stated sufficient range.
The pointwise all-length open gap gives a positive bound at each parameter. The preceding theorem preserves some positive bound on a neighborhood, uniformly in \(N\). A finite subcover of \(K\) and the minimum of its finitely many bounds give \(\delta {\gt}0\).
Let \(A_x^j\) be continuous normalized normal blocks, pairwise inequivalent under virtual similarity and phase at every parameter, with physical dimension \(d\geq 1\) and fixed bond dimensions \(D_j\). Put \(D_* =\max \{ \sum _jD_j,1\} \) and \(B_x=\bigoplus _j\mu _j(x)A_x^j\), with every weight nonzero. For every compact \(K\) and every fixed \(R\geq 3D_*^5\), one positive open norm gap holds uniformly for \(x\in K\) and all \(N\geq R\).
The dimension-bound span theorem gives, at each parameter, a positive simultaneous injectivity length \(L_x\) with \(L_x+1\leq 3D_*^5\). Propagation gives the common length \(S=3D_*^5-1{\gt}0\). Apply the compact open-gap theorem at \(R\geq S+1\).
Under the hypotheses of Theorem 12.8.1.35, let \(h_x\) be a continuous family of positive range-\(R\) interactions with \(\ker h_x=G_R(B_x)\). Then there is \(\delta {\gt}0\) such that \(\delta \| v\| \leq \| H^{\mathrm{open}}_N(h_x)v\| \) for every \(x\in K\), every \(N\geq R\), and every \(v\perp \ker H^{\mathrm{open}}_N(h_x)\). The same conclusion holds under the normalized normal-block hypotheses of Theorem 12.8.1.36 at \(R\geq 3D_*^5\). Nonzero weights need not be continuous.
The local ground projections are continuous at range \(R\). The canonical open gaps are uniform on \(K\) and all \(N\geq R\) by the two preceding compact results. Compact positive comparison gives one \(\kappa {\gt}0\) with \(\kappa h_R(B_x)\leq h_x\) on \(K\). Summing translates transfers the canonical gap with factor \(\kappa \). A pointwise upper comparison identifies the kernels.
Let \(A^0,\ldots ,A^{r-1}\) have positive physical and bond dimensions. Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Set \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\neq 0\), and fix \(R\geq S+1\). There are \(p{\gt}0\), with \(R\leq 2p\), and \(\delta {\gt}0\) such that
for every \(M\geq 2\) and every \(v\perp G_{pM}^{\mathrm{ES}}(B)\). This combines the block-injective local spaces of [ 17 , Section IV.C ] with the martingale estimate of [ 225 , Theorem 2.1(ii) and Section 6 ] .
Normalize the blocks while preserving all local spaces and the span at \(S\). The simultaneous span separates the blocks. Choose \(p\geq R\) and a uniform open-chain gap \(\gamma {\gt}0\) at range \(2p\) on every length \(pM\), \(M\geq 2\). The range-\(R\) and range-\(2p\) open kernels both equal \(G_{pM}^{\mathrm{ES}}\). On the interval of length \(2p\), finite-dimensional comparison gives \(\kappa h_{2p}\leq H^{\mathrm{open}}_{2p,R}\) with \(\kappa {\gt}0\). Open-range comparison transfers the gap to \(\delta =\kappa \gamma /(2p-R+1){\gt}0\) at range \(R\). Equality of all local spaces transports this bound to the original block sum.
Let \(A^0,\ldots ,A^{r-1}\) have positive physical and bond dimensions. Suppose that their simultaneous length-\(S\) word tuples span the product matrix algebra, where \(S{\gt}0\). Set \(B=\bigoplus _j\mu _jA^j\), with every \(\mu _j\neq 0\), and fix \(R\geq S+1\). There are \(m\geq R\) and \(\gamma {\gt}0\) such that \((R-1)^2{\lt}m\gamma \) and
for every \(v\perp G_{m+R-1}^{\mathrm{ES}}(B)\). The strict inequality is the finite-range threshold of Theorem 12.3.7.81; the coefficient is discussed in docs/paper-gaps/knabe88_finite_range_coefficient.tex.
Choose \(p{\gt}0\) and \(\delta {\gt}0\) giving the open-chain gap on all lengths \(pM\), \(M\geq 2\). Choose \(n\in \mathbb N\) with \((R-1)^2{\lt}n\delta \). Set \(M=n+2R+2\) and \(m=pM+1-R\). Then \(m\geq R\), \(n\leq m\), and \(m+R-1=pM\). Thus \(\gamma =\delta \) has both the required gap and \((R-1)^2{\lt}m\gamma \).
Let \(A^0,\ldots ,A^{r-1}\) have positive physical and bond dimensions. Suppose their word tuples of a supplied positive length \(S\) span the full product matrix algebra. Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA^j\), where every \(\mu _j\) is nonzero. For each interaction range \(R\geq S+1\), there is a constant \(\gamma _R{\gt}0\) such that
for every chain length \(N\in \mathbb N\) and every \(v\perp \ker H_{R,N}\), where \(H_{R,N}\) is the canonical periodic parent Hamiltonian of \(B\). This is the block-injective gap assertion of [ 17 , Section IV.C ] at the simultaneous injectivity range.
Normalize the blocks while retaining simultaneous injectivity at \(S\) and all local ground spaces. Simultaneous injectivity forces distinct blocks to be inequivalent. The grouped martingale estimate and finite-range periodic comparison provide a gapped interaction range \(W\geq 2R\). Let \(B'\) be the weighted sum of the normalized blocks. Trace preservation propagates simultaneous injectivity from \(S\) to every larger length, so, since \(R\geq S+1\), the open-chain ground-space theorem gives
Two-sided comparison of the periodic range-\(R\) and range-\(W\) Hamiltonians therefore transfers the gap to \(R\). Taking a minimum with the finitely many remaining finite-volume gaps makes the bound uniform over every \(N\). Finally, normalization left the original parent Hamiltonians unchanged.
Let \(A\) be a tensor in canonical form, with distinct normal representatives \(B_0,\ldots ,B_{g-1}\), and set \(B=\bigoplus _j B_j\). For every positive interaction range \(R\) and every chain length \(N\), the canonical periodic parent Hamiltonians satisfy
This equality includes the multiplicity copies and the ambient reconstruction of the original tensor.
For every positive length \(R\), the local spaces satisfy
Their orthogonal complementary projections therefore agree. Translating and summing those projections gives the same Hamiltonian.
Let \(A\) be a tensor in canonical form with positive physical dimension. Fix any canonical decomposition of \(A\), and let \(B_0,\ldots ,B_{g-1}\) be its distinct normal representatives. Let \(S\) be a positive integer such that the simultaneous length-\(S\) word tuples of \(B_0,\ldots ,B_{g-1}\) span the product of the block matrix algebras. Then, for every \(R\geq S+1\), there exists \(\gamma {\gt}0\) such that, for every chain length \(N\) and every vector \(v\perp \ker H_{R,N}(A)\),
Set \(B=\bigoplus _jB_j\). Its blocks satisfy the hypotheses of the uniform gap at the simultaneous injectivity range, so \(H_{R,N}(B)\) has a positive uniform gap. Since \(H_{R,N}(A)=H_{R,N}(B)\) for every \(N\), the same constant works for \(A\), without changing the range or the chain length.
Let \(A\) be a tensor in canonical form with positive physical dimension. One may choose a basis of normal tensors \(B_0,\ldots ,B_{g-1}\) such that the following holds for every positive integer \(S\): if the simultaneous length-\(S\) word tuples span the product of the block matrix algebras, then, for every \(R\geq S+1\), there exists \(\gamma {\gt}0\) such that, for every chain length \(N\) and every vector \(v\perp \ker H_{R,N}(A)\),
The representatives are chosen once, before \(S\) and \(R\). This includes canonical forms with repeated blocks, as in [ 17 , Section IV.C ] .
Fix one canonical decomposition of \(A\) and take its distinct normal representatives, which form a basis of normal tensors for \(A\). The preceding theorem applies to this fixed choice for every \(S\) and every \(R\geq S+1\).
Let \(A\) be a tensor in canonical form with positive physical dimension and positive bond dimension \(D\). For every \(R\geq 3D^5\), there is \(\gamma {\gt}0\) such that
for every \(N\in \mathbb N\) and every \(v\perp \ker H_{R,N}(A)\). The constant may depend on \(A\) and \(R\), but not on \(N\). This sufficient range combines the blocking estimate of [ 13 , lines 317–345 ] with the gap assertion of [ 17 , Section IV.C ] . It need not be optimal; the sharper sufficient condition \(R\geq S+1\) is recorded in [ 131 ] . The unrestricted assertion fails at some shorter ranges; see [ 27 ] .
Choose a canonical decomposition. The dimension estimate supplies a simultaneous injectivity length \(S{\gt}0\) for its distinct normal representatives, with \(S+1\leq 3D^5\leq R\). The canonical gap at that simultaneous injectivity length gives the assertion.
Let \(h\) be a positive interaction on \(R\) sites whose kernel is the local MPS space \(G_R(A)\), in the sense of [ 17 , Section IV.C ] . Write \(h_R=I-P_{G_R(A)}\) for the canonical parent projection. There are constants \(\kappa ,C{\gt}0\) such that
Apply the finite-dimensional comparison with the projection onto the orthogonal complement of the kernel of \(h\). This projection is \(h_R\).
Let \(h\) act on the physical space of \(R\) consecutive sites. For \(R\leq N\), let \(U_i\) be the unitary which places the cyclic interval of length \(R\) starting at \(i\) first and the remaining sites second. Define
When \(R{\gt}N\), set each translated interaction, and hence their sum, to zero. This agrees with the convention for the canonical parent Hamiltonian.
Cyclic extension and its periodic sum preserve operator inequalities and commute with scalar multiplication. In particular, for every volume \(N\),
The scalar identity holds for every complex number \(c\).
Extension by the identity preserves order and scalar multiplication, as does unitary conjugation. Summing over the cyclic positions preserves these relations. Positivity is the comparison with zero.
For a tensor \(A\) and a positive interaction range \(R\), the periodic sum of the canonical parent projection \(h_R=I-P_{G_R(A)}\) is exactly the canonical parent Hamiltonian, at every volume:
In cyclic active-interval coordinates, each translated canonical term is the full-interval parent projection extended by the identity on the spectator sites. Conjugating back identifies the terms individually. When \(R{\gt}N\), both definitions give zero.
Let \(R{\gt}0\) and let \(h\) be a fixed positive operator with kernel \(G_R(A)\), and let \(H_{R,N}(A)\) be the canonical periodic parent Hamiltonian. There are \(\kappa ,C{\gt}0\), independent of \(N\), such that
Theorem 12.8.1.45 gives \(\kappa ,C{\gt}0\) with \(\kappa h_R\leq h\leq Ch_R\). Each translated term \(h_i\) is \(h\) extended by the identity and conjugated by the unitary \(U_i\), so these inequalities hold term by term, \(\kappa (h_R)_i\leq h_i\leq C(h_R)_i\). Summing over the cyclic translates and using the monotonicity and homogeneity of the periodic sum from Theorem 12.8.1.47 gives
Theorem 12.8.1.48 identifies \(H_N(h_R)\) with \(H_{R,N}(A)\).
Let \(A\) be a matrix product tensor and let \(R{\gt}0\). Suppose that the periodic Hamiltonians formed from the orthogonal projection onto \(G_R(A)^\perp \) have a positive gap uniformly in the chain length. Let \(h\) be a fixed positive operator on \(R\) sites with kernel \(G_R(A)\), and let \(H_N(h)\) be the sum of its cyclic translates. Then there is \(\delta {\gt}0\) such that, for every chain length \(N\) and every \(v\perp \ker H_N(h)\),
For \(N{\lt}R\) both periodic Hamiltonians are defined to be zero.
Finite-dimensional positivity gives constants \(\kappa ,C{\gt}0\) such that \(\kappa h_R\leq h\leq C h_R\), where \(h_R=I-P_{G_R(A)}\) is the canonical parent projection. Cyclic extension and summation give the same inequalities for the periodic Hamiltonians, with constants independent of \(N\). The two inequalities imply equality of the kernels. Restriction to their common orthogonal complement therefore transfers a canonical gap \(\gamma \) to the gap \(\kappa \gamma \). This is the projector reduction of [ 17 , Section IV.C ] .
Let \(A\) be a tensor in canonical form with positive physical dimension. Fix any canonical decomposition of \(A\), and let \(B_0,\ldots ,B_{g-1}\) be its distinct normal representatives. Suppose that \(S{\gt}0\) and the tuples of length-\(S\) words span the product of the block matrix algebras. For every \(R\geq S+1\) and every fixed positive operator \(h\) on \(R\) sites with kernel \(G_R(A)\), there is \(\gamma {\gt}0\) such that, for every \(N\) and every \(v\perp \ker H_N(h)\),
Apply the canonical projector gap of Theorem 12.8.1.42 at range \(R\), then the comparison theorem for the fixed interaction \(h\). The constants depend on \(h\) but not on \(N\).
Let \(A\) be a tensor in canonical form with positive physical dimension. One may choose a basis of normal tensors \(B_0,\ldots ,B_{g-1}\) such that the following holds. Suppose that \(S{\gt}0\) and the tuples of length-\(S\) words span the product of the block matrix algebras. For every \(R\geq S+1\) and every fixed positive operator \(h\) on \(R\) sites with kernel \(G_R(A)\), there is \(\gamma {\gt}0\) such that, for every \(N\) and every \(v\perp \ker H_N(h)\),
The representatives are chosen once, independently of \(S\), \(R\), and \(h\).
Apply the canonical projector gap at range \(R\), then the comparison theorem for the fixed interaction \(h\). The constants depend on \(h\) but not on \(N\).
Let \(A\) be a tensor in canonical form with positive physical dimension and positive bond dimension \(D\). Let \(R\geq 3D^5\) and let \(h\) be any fixed positive operator on \(R\) sites with \(\ker h=G_R(A)\). Then there is \(\gamma {\gt}0\) such that
for every \(N\in \mathbb N\) and every \(v\perp \ker H_N(h)\). The constant may depend on \(A\), \(R\), and \(h\), but not on \(N\). This is the positive-interaction comparison of [ 17 , Section IV.C ] at the sufficient dimension bound of [ 13 , lines 317–345 ] .
The canonical projector interaction has a uniform positive periodic gap at range \(R\). Comparison with the fixed interaction \(h\) preserves the kernel and gives a positive gap independent of \(N\).
12.9 Prescribed gaps at sufficiently long interaction ranges
Write \(H_{R,N}\) for the periodic parent Hamiltonian on \(N\) sites with interaction range \(R\). The local interactions in this section are orthogonal projectors with coefficient one. The range increases with the requested gap; no rescaling of the Hamiltonian is made. The bounds follow from the projector-decay and martingale estimates of [ 225 , Theorem 2.1 and Section 6 ] and the finite-range comparison. They do not improve the gap at a prescribed shorter interaction range.
Let \(0{\lt}\eta {\lt}1\), \(1\le R\le m\), and \(2(R-1)^2\le m\eta \). Suppose the open parent Hamiltonian on \(m+R-1\) sites has gap at least \(1-\eta /2\). Then, for every \(N\ge 2m\) and every \(v\perp \ker H_{R,N}\),
Put \(l=R-1\) and \(\gamma =1-\eta /2\). The finite-range comparison gives \(\delta =(m\gamma -l^2)/(m-l)\). Its denominator is positive, and
Thus \(\delta \ge 1-\eta {\gt}0\), which also verifies the positive-numerator hypothesis of the comparison.
Let \(R{\gt}0\). If the range-\(R\) parent Hamiltonian on \(n\) sites has norm gap \(\kappa {\gt}0\) on its kernel complement, its extension to the prefix of an \((n+q)\)-site chain has the same lower bound. Likewise, if \(R\leq W\leq n\leq N\) and the range-\(R\) Hamiltonian on \(W\) sites has gap at least \(\kappa \), its interaction sum on \([n-W,n)\), acting on the \(N\)-site chain, has gap at least \(\kappa \). The physical coordinate isometry identifies this interval interaction sum with the \(W\)-site Hamiltonian tensored with the identity on the remaining sites. These are the finite-interval gaps used in condition C2 of [ 225 ] .
For the finite-interval operator \(T\) and its ground projection \(P\), the supplied gap gives \(\kappa (I-P)\leq T\). Extension by free sites preserves this inequality. The kernel projection extends to \(P\otimes I\), and the physical coordinate isometry transports both the Hamiltonian and this projection to the chosen interval. Thus the extended operator dominates \(\kappa \) times its excitation projection. On the kernel complement, the quadratic-form bound and Cauchy–Schwarz give the asserted norm bound.
Let \(H\) be a linear operator on a finite-dimensional complex Hilbert space, let \(W=\ker H\), and suppose \(W\subseteq U\cap V\). Write \(P_W,P_U,P_V\) for the orthogonal projections onto these subspaces. Suppose \(\kappa \geq 0\), \(c{\lt}1\), \(\| P_W-P_UP_V\| \leq c\), and \(\kappa ((I-P_U)+(I-P_V))\leq H\) in the order of self-adjoint operators. Then \(\kappa (1-c)\| v\| \leq \| Hv\| \) for every \(v\perp W\). This is a two-subspace consequence of the ground-projection estimate in [ 225 , Section 6, the three-interval commutation lemma, part (ii) ] .
If \(z\in U\cap V\) and \(z\perp W\), the projection defect gives \(\| z\| \leq c\| z\| \), hence \(z=0\). Subtracting \(P_Wv\) from an arbitrary \(v\in U\cap V\) proves \(W=U\cap V\). Put \(L=(I-P_U)+(I-P_V)\). The projection defect bounds the Friedrichs overlap by \(c\). The complementary-projection anticommutator inequality gives \(L^2\geq (1-c)L\), and \(\ker L=U\cap V=W\). The finite-dimensional spectral theorem gives \((1-c)\| v\| ^2\leq \langle Lv,v\rangle \) for \(v\perp W\). Applying \(\kappa L\leq H\) and Cauchy–Schwarz yields \(\kappa (1-c)\| v\| ^2\leq \langle Hv,v\rangle \leq \| Hv\| \| v\| \). Cancel \(\| v\| \) when \(v\ne 0\).
Let \(H\) be the range-\(R\) parent Hamiltonian on an open chain of length \(N\), and let \(H_L\) and \(H_T\) be its interaction sums on a prefix of length \(n\) and on a terminal interval of length \(W\), respectively. Write \(P_L\), \(P_T\), and \(P\) for their orthogonal ground projections. Suppose \(\kappa \geq 0\), and suppose both interval Hamiltonians have a norm gap at least \(\kappa \) on their respective kernel complements. If \(\| P-P_LP_T\| \leq 1/2\), then \((\kappa /16)\| v\| \leq \| Hv\| \) for every \(v\perp \ker H\). The interval interaction sums always satisfy \(H_L\leq H\) and \(H_T\leq H\).
Each interval interaction sum is a subfamily of the positive terms of \(H\), so \(H_L+H_T\leq 2H\). The interval gaps give \(\kappa (P_L^\perp +P_T^\perp )\leq H_L+H_T\). For \(v\perp \ker H\), the projection defect gives \(\| P_LP_Tv\| \leq \| v\| /2\). Decomposing \(v=P_LP_Tv+P_LP_T^\perp v+P_L^\perp v\) yields \(\| v\| \leq 2(\| P_L^\perp v\| +\| P_T^\perp v\| )\). Squaring gives \(\| v\| ^2\leq 8(\| P_L^\perp v\| ^2+\| P_T^\perp v\| ^2)\). Hence \(\kappa \| v\| ^2\leq 16\langle Hv,v\rangle \). Cauchy–Schwarz and cancellation of \(\| v\| \) prove the claim.
Let \(A\) be a normalized primitive tensor with positive bond dimension and a positive-definite invariant matrix. For every \(a{\gt}0\), every sufficiently large integer \(l\) satisfies \(l{\gt}1\), \(l\)-block injectivity, and a number \(\epsilon _l\ge 0\) with \(\epsilon _l\sqrt{l+1}{\lt}a\) such that, for every prefix length \(K\ge 0\),
Here \(G_I\) denotes the projection onto the matrix-product boundary space on the indicated interval, extended by the identity on its complement in the chain of length \(K+l+1\).
The geometric estimate gives \(\epsilon _l=c\lambda ^l(1+c\lambda ^l)/(1-c\lambda ^l)\) beyond the injectivity threshold, with \(c{\gt}0\) and \(0{\lt}\lambda {\lt}1\). Since \(\sqrt{l+1}\lambda ^l\to 0\) and \(c\lambda ^l\to 0\),
Choose \(l\) beyond the injectivity threshold, above one, and sufficiently large that \(c\lambda ^l{\lt}1\) and this product is below \(a\).
Let \(A\) be a normalized primitive tensor with positive bond dimension and a positive-definite invariant matrix. For every \(0{\lt}\eta {\lt}1\), every sufficiently large integer \(l\) has the following property. If \(m\ge l+1\), \(2l^2\le m\eta \), and \(N\ge 2m\), then
for every \(v\perp \ker H_{l+1,N}\). One may take \(m=\max \{ l+1,\lceil 2l^2/\eta \rceil \} \). The threshold for \(l\) depends on the tensor; no dimension-only bound on this threshold is asserted.
The geometric projector estimate gives \(\epsilon _l=c\lambda ^l(1+c\lambda ^l)/(1-c\lambda ^l)\) and \(\epsilon _l\sqrt{l+1}\to 0\). Choose \(l\) above the injectivity threshold and sufficiently large that \(0\le s_l=\epsilon _l\sqrt{l+1}{\lt}\eta /4\). The open-chain martingale coefficient satisfies
Apply Theorem 12.9.1 with \(R=l+1\).
Let \(B=\bigoplus _j\mu _jA_j\), where the \(A_j\) are normalized primitive tensors with common positive physical dimension, positive bond dimensions, and positive-definite invariant matrices. Suppose \(\mu _j\ne 0\) and no two blocks are related by similarity and multiplication by any nonzero complex scalar. For every \(0{\lt}\eta {\lt}1\) and every sufficiently large integer \(p\), the periodic range-\(2p\) parent Hamiltonian has gap at least \(1-\eta \) at every sufficiently large chain length.
Choose \(p\) above the open-kernel threshold and large enough that the three-interval projector error is at most \(\epsilon =\eta /(4\sqrt2)\). The grouped open-chain bound on \(pM\) sites has coefficient \((1-\epsilon \sqrt2)^2=(1-\eta /4)^2\ge 1-\eta /2\). Choose \(M\) large enough that \(m=pM-2p+1\) satisfies \(2p\le m\) and \(2(2p-1)^2\le m\eta \). The active comparison window has \(m+2p-1=pM\) sites, so Theorem 12.9.1 applies for every \(N\ge 2m\).
The conclusion of Theorem 12.9.6 holds for every normal tensor of positive bond dimension, without a normalization hypothesis. Likewise, the conclusion of Theorem 12.9.7 holds for normal blocks with nonzero weights and positive physical and bond dimensions, provided no two blocks are related by similarity and multiplication by any nonzero complex scalar.
Replace the tensor, or each block independently, by its normalized primitive representative. The canonical parent Hamiltonians agree exactly at every interaction range and every chain length, so the prescribed lower bounds transfer unchanged.
The direct sum of the three bond-one qubit tensors \((1,0)\), \((0,1)\) and \((1/\sqrt2,1/\sqrt2)\), with weights one, has a basis of three normalized, normal, pairwise inequivalent blocks. Its simultaneous injectivity length is exactly two. Its two-site local ground space is \(\operatorname {Sym}^2(\mathbb C^2)\), and its canonical parent interaction is the singlet projection \(h_2=(I-F)/2\), where \(F\) exchanges the two physical sites.
The word tuples for \(00\), \(11\) and \(01\) are \((1,0,1/2)\), \((0,1,1/2)\) and \((0,0,1/2)\), which span \(\mathbb C^3\); two one-site words cannot span this space. Each scalar block has transfer normalization one. Simultaneous spanning separates the blocks and gives eventual independence of their periodic vectors. The two-site boundary vectors span \(|00\rangle \), \(|11\rangle \) and \(|++\rangle \), exactly the symmetric subspace. Its orthogonal complement is the singlet line.
For the preceding tensor, the periodic range-two parent Hamiltonian has a nonzero eigenvector of eigenvalue \(\varepsilon _N=1-\cos (2\pi /N){\gt}0\) for every \(N\geq 2\). These eigenvalues tend to zero. Consequently there are no \(\delta {\gt}0\) and \(N_0\) such that
Thus the unrestricted assertion in [ 17 , Section IV.C, lines 2183–2187 ] is false. The sufficient-range theorems remain valid, since this example has \(R=S=2\). See [ 27 ] .
On configurations with one down spin the sum of singlet projections acts as \(f(j)-(f(j+1)+f(j-1))/2\). The first Fourier character of \(\mathbb Z/N\mathbb Z\) is therefore an eigenvector with the stated eigenvalue. Its coefficient at the zero site is one, so it is nonzero. Self-adjointness makes it orthogonal to the kernel. A uniform positive norm bound would give \(\delta \leq \varepsilon _N\) for every sufficiently large \(N\), contradicting \(\varepsilon _N\to 0\).
Let \(T:E\to F\) be an injective bounded linear map between real or complex Hilbert spaces, with \(E\) finite-dimensional. Write \(K=(T^*T)^{-1}\) and let \(P\) be the orthogonal projection onto \(\operatorname {ran}T\). For every bounded operator \(X\) on \(F\) and every scalar \(c\),
For a family \(T_i,X_i,c_i\) indexed by an arbitrary filter, suppose \(T_i\) is eventually injective and its inverse Gram operators are eventually bounded by one constant. Then \(\| T_i^*X_iT_i-c_iT_i^*T_i\| \to 0\) implies \(\| P_iX_iP_i-c_iP_i\| \to 0\). The Hilbert spaces may vary with \(i\). For any finite-dimensional ground subspace \(G\) and unit vector \(v\in G\), with orthogonal projection \(P_G\), one also has
Since \(P=TKT^*\), one has \(PXP=(TK)(T^*XT)(KT^*)\). Both outside factors have norm \(\sqrt{\| K\| }\). Their product therefore gives the first bound. Apply it to \(X-cI\) for the second. The limiting assertion follows by bounding the compression defect by a fixed constant times the pullback defect. The unit-vector bounds follow from \(P_Gv=v\) and the Cauchy–Schwarz inequality.
Place a \(k\)-site observable \(X\) between free intervals of lengths \(\ell ,r\), obtaining \(X^{(\ell ,r)}=I_\ell \otimes X\otimes I_r\). Placement preserves products and adjoints. When \(k\geq 1\), it is the usual nonwrapping window inclusion and \(\| X^{(\ell ,r)}\| \leq \| X\| \).
Tensor products multiply factorwise and commute with adjoints. The nonwrapping window inclusion is a unital homomorphism preserving adjoints between matrix algebras, so it is contractive.
Let \(H\geq 0\) act on a finite-dimensional complex Hilbert space, let \(K=\ker H\), and suppose \(\gamma \geq 0\) and \(\gamma \| v\| \leq \| Hv\| \) for every \(v\perp K\). Then \(\langle Hv,v\rangle \geq \gamma (\| v\| ^2-\| P_Kv\| ^2)\) for every \(v\). For every \(\psi \in K\) and linear operator \(X\),
If \(K=\mathbb {C}\psi \) and \(\| \psi \| =1\), the subtracted term is \(|\langle \psi ,X\psi \rangle |^2\). This is the finite-volume estimate for the limiting argument in [ 225 , Section 6 ] ; the full ground-space projection is retained as described in [ 61 ] .
The gap gives \(H\geq \gamma P_{K^\perp }\). Pythagoras yields \(\| P_{K^\perp }v\| ^2=\| v\| ^2-\| P_Kv\| ^2\). Since \(H\psi =0\), the adjoint identity gives \(\langle \psi ,X^*[H,X]\psi \rangle =\langle HX\psi ,X\psi \rangle \). Apply the first estimate to \(v=X\psi \). For a one-dimensional ground space, its projection is \(P_Kv=\langle \psi ,v\rangle \psi \).
Let \(H_n,X_n\) act on finite-dimensional complex Hilbert spaces which may depend on \(n\). Suppose that, for all sufficiently large \(n\), \(H_n\geq 0\), \(\psi _n\in \ker H_n\), and \(H_n\) has norm gap at least the same constant \(\gamma \geq 0\) on \((\ker H_n)^\perp \). Let \(a,e\in \mathbb {R}\). If \(\| X_n\psi _n\| ^2\to a\), \(\| P_{\ker H_n}X_n\psi _n\| ^2\to 0\), and \(\langle \psi _n,X_n^*[H_n,X_n]\psi _n\rangle \to e\), then \(\gamma a\leq e\).
For all sufficiently large \(n\), (410) gives the required inequality before taking limits. Its left side tends to \(\gamma (a-0)=\gamma a\) and its right side tends to \(e\). The order relation on \(\mathbb {R}\) is closed.
Let \(H_n,X_n\) act on finite-dimensional complex Hilbert spaces which may depend on \(n\). Suppose that, for all sufficiently large \(n\), \(H_n\geq 0\), \(\| \psi _n\| =1\), \(\ker H_n=\mathbb {C}\psi _n\), and \(H_n\) has norm gap at least the same constant \(\gamma \geq 0\) on \((\ker H_n)^\perp \). If \(\| X_n\psi _n\| ^2\to a\), \(\langle \psi _n,X_n\psi _n\rangle \to 0\), and \(\langle \psi _n,X_n^*[H_n,X_n]\psi _n\rangle \to e\), then \(\gamma a\leq e\). Finite-volume uniqueness is a hypothesis of this corollary.
The rank-one projection formula makes the subtracted term the squared absolute value of the mean. This tends to zero, so the finite-volume estimate passes to the limit.
Let \(p\geq 1\), \(M\geq 2\), and \(pM\leq N\). Sample the canonical range-\(2p\) interactions at starts \(0,p,\ldots ,p(M-2)\), and write \(H_{\mathrm{sp}}\) for their sum on \(N\) sites. Let \(U\) be the kernel of the full range-\(2p\) interaction on the prefix of length \(pM\). Suppose the grouped martingale projections satisfy \(\| Q_nE_n\| \leq \epsilon {\lt}1/\sqrt2\), with \(\epsilon \geq 0\). Then
The grouped projector estimate remains valid after adjoining free sites on the right. If \(N=pM+q\) with \(q{\gt}0\), adjoining the terminal range-\(2p\) parent projection \(T\) gives \(H_{\mathrm{sp}}+T\leq H^{\mathrm{open}}_{N,2p}\). This is a consequence of [ 225 , Theorem 2.1(ii) ] .
Apply the grouped martingale energy inequality directly to the sampled interaction sum. Every sampled term annihilates \(U\), so decomposition into \(U\) and \(U^\perp \) gives the operator inequality. Free-site extension does not increase the projector norm. When \(q{\gt}0\), the terminal start \(N-2p=p(M-2)+q\) is strictly larger than all sampled starts. Thus the sampled terms and the terminal term form a subfamily of the full positive interaction sum, each counted once.
Let \(p\geq 1\), \(M\geq 2\), \(0{\lt}q{\lt}p\), and \(N=pM+q\). Assume that every range-\(2p\) open Hamiltonian at length at least \(2p\) has its boundary-condition space as kernel. Suppose that the three-interval ground-projection defect is at most \(0\leq \epsilon {\lt}1/\sqrt2\) whenever the overlap has at least \(p\) sites, uniformly in both outer intervals, with the right interval nonempty. Then
Let \(U\) be the prefix ground space at length \(pM\), and let \(V\) be the terminal ground space on \(2p\) sites. Their overlap contains \(2p-q\geq p\) sites. Put \(\kappa =(1-\sqrt2\epsilon )^2\leq 1\). The sparse energy bound and the terminal projection give \(H^{\mathrm{open}}_{N,2p}\geq \kappa ((I-P_U)+(I-P_V))\). The full ground space is contained in both \(U\) and \(V\), and the three-interval estimate bounds its projection defect by \(\epsilon \). The sharp two-ground-space comparison gives the conclusion.
Let \(A_0,\ldots ,A_{r-1}\) be normalized primitive tensors with common physical dimension \(d\geq 1\), positive bond dimensions, and positive-definite invariant matrices. Suppose no two distinct blocks are related by a virtual similarity and a phase. Put \(B=\bigoplus _j\mu _jA_j\), with \(\mu _j\ne 0\) for every \(j\). For every \(0{\lt}\eta {\lt}1\), there is \(p_0\) such that, for every \(p\geq p_0\) and every \(N\geq 2p\),
This is a quantitative consequence of [ 225 , Theorem 2.1(ii) and Section 6 ] .
Choose \(\epsilon =\eta /(4\sqrt2)\) and take \(p\) above the kernel threshold and the projector-decay threshold for every overlap length at least \(p\). Write \(N=pM+q\), with \(M\geq 2\) and \(0\leq q{\lt}p\). For \(q=0\), the grouped gap is \((1-\sqrt2\epsilon )^2\). For \(q{\gt}0\), the residual-interval theorem gives \((1-\sqrt2\epsilon )^2(1-\epsilon )\). Both coefficients are at least \(1-\eta \).
Let \(A_0,\ldots ,A_{r-1}\) be normal tensors with common physical dimension \(d\geq 1\) and positive bond dimensions. Suppose no two distinct blocks are related by similarity and multiplication by a nonzero complex scalar, and let \(B=\bigoplus _j\mu _jA_j\), with \(\mu _j\ne 0\) for every \(j\). For every \(0{\lt}\eta {\lt}1\), there is \(p_0\) such that, for every \(p\geq p_0\) and every \(N\geq 2p\),
Thus the canonical open-chain gap approaches one as the even interaction range grows, uniformly over all lengths at least that range. This is a quantitative consequence of the projector decay and martingale estimates of [ 225 , Theorem 2.1(ii) and Section 6 ] .
Independent primitive normalization preserves all block ground spaces and therefore every assembled canonical open Hamiltonian. It also preserves inequivalence. Apply the primitive block theorem to the normalized representatives and transfer its bound through the exact equality of open Hamiltonians.
For a tensor \(A\), a matrix \(\rho \) with nonzero trace, and an observable \(X\) on \(k\) sites, write \(E_X(Y)=\sum _{\sigma ,\tau }X_{\sigma \tau }A^\tau Y(A^\sigma )^*\) and \(\eta _A(X)=\operatorname {tr}(E_X(\rho ))/\operatorname {tr}\rho \). This is the local expectation appearing in [ 225 , Section 6 ] .
For an observable \(X\) on \(k\) sites and nonnegative integers \(\ell ,r\), let \(X_{\ell ,r}=I_\ell \otimes X\otimes I_r\) on \(N=\ell +k+r\) sites, with the tensor factors ordered from left to right.
Let \(\Gamma _n\) be the Hilbert-space boundary map and let \(\mathcal R\) denote Gram reshuffling. Then
Pair both sides of the first identity with matrix-unit boundary vectors. The resulting finite sums agree term by term. Insertion transfers multiply under consecutive tensor products, while the identity insertion on \(n\) sites is \(E^n\). These facts give the remaining identities.
Let \(A\) be a normalized primitive tensor with invariant matrix \(\rho \), and let \(P_\rho (Y)=\operatorname {tr}(Y)\rho /\operatorname {tr}\rho \). For every local observable \(X\), \(P_\rho E_XP_\rho =\eta _A(X)P_\rho \) and \(E^n\to P_\rho \) in operator norm. Along any filter on which \(\ell ,r\to \infty \), \(E^\ell E_XE^r\to \eta _A(X)P_\rho \).
Substitution into the rank-one projection gives the first identity. For \(n\geq 1\), the decomposition \(E^n=P_\rho +(E-P_\rho )^n\) and decay of the complementary powers give the second. Continuity of multiplication gives the insertion limit.
Under the preceding hypotheses, with \(K_\infty =\mathcal R(P_\rho )\), \(\Gamma _N^*X_{\ell ,r}\Gamma _N\to \eta _A(X)K_\infty \) as both free intervals tend to infinity.
Apply the continuous linear reshuffling map to the insertion limit.
Suppose in addition that \(\rho \) is positive definite. Let \(G_N\) be the open-boundary MPS space and \(P_N\) its orthogonal projection. Then
In particular, centered observables satisfy \(\| P_NX_{\ell ,r}P_N\| \to 0\). For every \(\varepsilon {\gt}0\), eventually \(\| P_NX_{\ell ,r}\psi -\eta _A(X)\psi \| {\lt}\varepsilon \) simultaneously for all \(\psi \in G_N\) with \(\| \psi \| \leq 1\).
Subtract \(\eta _A(X)\Gamma _N^*\Gamma _N\) from the limiting boundary compression. The difference tends to zero. The boundary maps are eventually injective and their inverse Gram operators remain bounded. The compression estimate therefore gives the operator-norm limit. Apply the resulting operator to vectors of norm at most one.
With the preceding hypotheses, let \(\psi _N\) be any varying family of vectors eventually belonging to \(G_N\). If \(\| \psi _N\| \leq 1\) and \(\eta _A(X)=0\), then \(\| P_NX_{\ell ,r}\psi _N\| ^2\to 0\). If instead \(\| \psi _N\| =1\), then \(\langle \psi _N,X_{\ell ,r}\psi _N\rangle \to \eta _A(X)\). These assertions concern one primitive sector; the general pure-GVBS infinite-volume statement is discussed in [ 61 ] .
The projection norm is bounded by the centered compression norm. Squaring preserves its zero limit. For unit vectors, the scalar expectation deviation is bounded by the scalar compression norm.
For any tensor \(A\), positive observable \(X\) on \(k\) sites, and positive virtual matrix \(\rho \), the insertion \(E_X(\rho )\) is positive. This is the positivity of the local GVBS contraction in [ 225 , Section 3, equations (3.1)–(3.2b) ] .
Let \(B_{a,(\sigma ,b)}=(A^\sigma )_{ab}\), where \(\sigma \) runs over words of length \(k\). The insertion is \(E_X(\rho )=B(X^{\mathsf T}\otimes \rho )B^*\). Transposition, tensor products, and congruence preserve positivity.
Fix \(A\) and a virtual matrix \(\rho \) with nonzero trace. For each interval length \(k\), the functional \(\eta _A\) is complex linear. If \(\rho \geq 0\), then \(\eta _A(X)\geq 0\) for \(X\geq 0\). If \(E(\rho )=\rho \), then \(\eta _A(I_k)=1\) and adjoining identity factors on the right preserves expectations. If \(\sum _i(A^i)^*A^i=I\), adjoining identity factors on the left preserves expectations. Thus a normalized primitive tensor with its positive invariant matrix defines compatible states on finite intervals, as in [ 225 , Section 3, equations (3.1)–(3.2b) ] .
Linearity follows from linearity of the insertion and the trace. The positive numerator and denominator give positivity. At a fixed point, \(E^m(\rho )=\rho \), so the identity insertion \(E_{I_m}=E^m\) gives normalization. Consecutive insertions compose, giving \(E_{X\otimes I_m}=E_XE^m\) and \(E_{I_m\otimes X}=E^mE_X\). Invariance removes the right factor, while trace preservation removes the left factor after taking the trace. The primitive assumptions include positivity, invariance, nonzero trace, and trace preservation.
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \). Let \(X\) act on \(k\) sites and let \(\ell ,r\to \infty \) along any filter. For any family of unit vectors \(\psi _N\) eventually belonging to the open-boundary MPS space on \(N=\ell +k+r\) sites, \(\| X_{\ell ,r}\psi _N\| ^2\to \operatorname {Re}\eta _A(X^*X)\).
The interior inclusion preserves products and adjoints, so \(\langle \psi _N,(X^*X)_{\ell ,r}\psi _N\rangle =\| X_{\ell ,r}\psi _N\| ^2\). Apply local expectation convergence to \(X^*X\) and take real parts.
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \), and let \(X\) be a fixed observable with \(\eta _A(X)=0\). Consider a sequence \(N_n=\ell _n+k+r_n\) with both \(\ell _n,r_n\to \infty \). Suppose the operators \(H_n\) are eventually positive, have kernel equal to the open-boundary MPS space, and have a common norm gap \(\gamma \geq 0\) on its orthogonal complement. Let \(\psi _n\) be unit vectors eventually in that ground space. If
then \(\gamma \operatorname {Re}\eta _A(X^*X)\leq e\). The energy limit is an explicit hypothesis here; the passage to the infinite-volume state in [ 225 , Section 6 ] remains separate, as discussed in [ 61 ] .
The excitation norm squared tends to \(\operatorname {Re}\eta _A(X^*X)\), while the squared norm of its full ground-space projection tends to zero. Take limits in the finite-volume commutator gap inequality.
With \(I,J\) as above, \(C_{a-b,b+N+c}(\iota _{I,J}(X))=I_b\otimes C_{a,N}(X)\otimes I_c\). Thus the integer-interval inclusion agrees with the interior placement used in the local GVBS expectations of [ 225 , Section 3, equations (3.1)–(3.2b) ] .
Write \(\sigma =(\sigma _1,\sigma _2,\sigma _3)\) and \(\tau =(\tau _1,\tau _2,\tau _3)\) for the restrictions of configurations to the first \(b\) sites, the middle \(N\) sites, and the last \(c\) sites. The interval inclusion formula gives
which is the entry of \(I_b\otimes C_{a,N}(X)\otimes I_c\).
Let \(\rho \geq 0\) satisfy \(E_A(\rho )=\rho \) and \(\operatorname {tr}\rho \neq 0\). For each finite interval, the insertion expectation \(\eta _A\) is a positive continuous linear functional of operator norm one. Consequently
The adjoint identity requires only \(\rho \geq 0\); the norm-one assertion also uses invariance and nonzero trace. Primitivity is unnecessary. This is the finite local state of [ 225 , Section 3, equations (3.1)–(3.2b) ] .
Positivity and complex linearity give a positive functional on the finite matrix algebra. Its norm is its value at the identity, which is one. A positive complex-linear functional preserves adjoints.
For an arbitrary range-\(R\) matrix \(h\), let \(H_N(h)=\sum _{a=0}^{N-R}\iota _a(h)\), where \(\iota _a\) places \(h\) on the sites \(a,\ldots ,a+R-1\) of an \(N\)-site chain, with the sum empty when \(N{\lt}R\). For \(R\geq 1\) and \(N\geq R\), its Euclidean matrix action is the open interaction Hamiltonian. If \(h\) is the complement projection of \(G_R(A)\), it is the canonical open parent Hamiltonian. This is the nonwrapping interaction sum in [ 225 , equation (3.12) ] .
The admissible starts are exactly \(0,\ldots ,N-R\). Each matrix inclusion acts as the corresponding Euclidean local interaction, so their sums agree. Substitute the local complement projection for the canonical case.
Observables supported on two disjoint nonwrapping intervals of a finite chain commute. No self-adjointness assumption is required.
Operators on the two tensor factors commute; each acts as the identity on the support of the other.
Let \(R,k\geq 1\), put \(t=R-1\) and \(w=2t+k\), and let \(X\) be supported on \(k\) sites. On \(w\) sites set \(B=I_t\otimes X\otimes I_t\) and
Neither positivity nor self-adjointness of \(h\) is needed for this definition. It is the fixed local expression used in the limiting argument of [ 225 , Section 6 ] .
Let \(R,k\geq 1\) and \(X_N=I_\ell \otimes X\otimes I_r\), where \(\ell ,r\geq t=R-1\) and \(N=\ell +k+r\). Then
Equivalently, both identities hold for an arbitrary placement of the enlarged interval inside a longer chain. These are locality identities; no spectral-gap assumption is made.
Terms disjoint from the support of \(X\) have zero commutator. The remaining terms are precisely the translates in the enlarged interval. Reindex their sum and use multiplicativity of the interval inclusion. Multiplying by \(X_N^\dagger \) gives the second identity.
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \), let \(h\) act on \(R{\gt}0\) sites, and let \(X\) act on \(k{\gt}0\) sites. Put \(t=R-1\), \(w=2t+k\), \(B=I_t\otimes X\otimes I_t\), and \(Y=B^*[H_w(h),B]\). On chains of length \(N_n=2n+w\), place \(X\) at start \(n+t\). For any eventually unit vectors \(\psi _n\) in the open-boundary MPS space, \(\operatorname {Re}\langle X_n^*[H_{N_n}(h),X_n]\psi _n,\psi _n\rangle \to \operatorname {Re}\eta _A(Y)\).
The commutator energy operator is the inclusion of \(Y\) starting at \(n\). Apply the primitive local expectation limit to \(Y\). Conjugate symmetry of the inner product preserves its real part.
Let \(A\) and \(\rho \) have the preceding properties. Along any filter with \(N_i\to \infty \), there exist vectors \(\psi _i\) in the open-boundary MPS space at every length \(N_i\), whose norms are eventually one.
Choose a nonzero matrix-unit boundary vector \(u\) and set \(\psi _i=\| \Gamma _{N_i}u\| ^{-1}\Gamma _{N_i}u\), with zero inverse interpreted as zero. The boundary map is eventually injective, so these vectors are eventually normalized; their ground-space membership holds at every length.
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \), and let \(X\) act on \(k{\gt}0\) sites with \(\eta _A(X)=0\). Fix \(R{\gt}0\) and let \(h=I-P_{G_R(A)}\). Suppose that, on the expanding chains \(N_n=2n+2(R-1)+k\), the canonical open Hamiltonians eventually have kernel \(G_{N_n}(A)\) and a common norm gap \(\gamma \geq 0\). Suppose also that a family of unit vectors eventually belongs to these ground spaces. With \(B=I_{R-1}\otimes X\otimes I_{R-1}\),
This is a statement about finite-interval primitive-sector expectations. Their extension to the quasi-local algebra and the GNS Hamiltonian remains separate, as discussed in [ 61 ] .
The excitation norm squared tends to \(\operatorname {Re}\eta _A(X^*X)\); its full ground-space projection tends to zero. The canonical matrix interaction sum is the open parent Hamiltonian. Apply the commutator energy limit and take limits in the finite-volume gap inequality.
Let \(A\) be a normalized primitive tensor of bond dimension \(D{\gt}0\) with positive-definite invariant matrix \(\rho \). At each fixed range \(R\geq D^4+1\), there exists \(\gamma {\gt}0\) such that the preceding local commutator inequality holds for every observable on every nonempty finite interval with \(\eta _A(X)=0\). The interaction is \(h=I-P_{G_R(A)}\).
Primitivity and positive definiteness imply normality. The dimension bound gives injectivity at \(D^4\) and hence at \(R-1\). The open-chain kernel is therefore the MPS ground space, and the normal open-chain gap theorem supplies one positive constant independent of length. Choose eventually normalized boundary vectors and apply the preceding inequality.
Let \(h\geq 0\) act on \(R{\gt}0\) sites with kernel \(G_R(A)\), and let \(q=I-P_{G_R(A)}\). For every chain length \(N\), \(\ker H_N(h)=\ker H_N(q)\).
There are \(\kappa ,C{\gt}0\) with \(\kappa H_N(q)\leq H_N(h)\leq C H_N(q)\) at every length. For positive operators, either bound gives the corresponding reverse inclusion of kernels. Their combination gives equality.
Let \(A\) be a normalized primitive tensor with positive-definite invariant matrix \(\rho \), let \(h\geq 0\) act on \(R{\gt}0\) sites, and let \(X\) act on \(k{\gt}0\) sites with \(\eta _A(X)=0\). Suppose the open interaction sums on chains \(N_n=2n+2(R-1)+k\) eventually have kernel \(G_{N_n}(A)\) and common norm gap \(\gamma \geq 0\). Suppose that a family of unit vectors eventually belongs to these ground spaces. With \(B=I_{R-1}\otimes X\otimes I_{R-1}\),
The open matrix sum represents the open interaction Hamiltonian. Its commutator energy tends to the expectation of the fixed local patch. Apply the primitive boundary commutator reduction, which derives the excitation norm and full ground-space projection limits from the tensor.
Let \(A\) be a normalized primitive tensor of bond dimension \(D{\gt}0\) with positive-definite invariant matrix \(\rho \). Let \(h\geq 0\) act on \(R\geq D^4+1\) sites with kernel \(G_R(A)\). There exists \(\gamma {\gt}0\) such that the preceding local commutator bound holds for every observable on every nonempty finite interval with \(\eta _A(X)=0\). This concerns one primitive sector and its finite-interval expectations; the quasi-local, GNS, and pure multiblock conclusions remain separate [ 61 ] .
The canonical open Hamiltonian has a uniform positive gap and the MPS ground-space kernel. Positive local comparison transfers the gap and preserves that kernel. Choose eventually normalized boundary vectors and apply the preceding local commutator inequality.
Let \(T\colon E\to F\) and \(S\colon G\to F\) be injective bounded linear maps between real or complex Hilbert spaces, with \(E\) and \(G\) finite-dimensional. Write \(K_T=(T^\dagger T)^{-1}\), \(K_S=(S^\dagger S)^{-1}\), and let \(P_T,P_S\) be the orthogonal projections onto their ranges. For every bounded operator \(X\) on \(F\),
Consequently, along any filter, if these maps are eventually injective, their inverse Gram norms are eventually bounded by constants, and \(\lVert T^\dagger X S\rVert \to 0\), then \(\lVert P_T X P_S\rVert \to 0\). The Hilbert spaces and maps may vary with the index.
Since \(P_T=TK_TT^\dagger \) and \(P_S=SK_SS^\dagger \), factor the compression as \((TK_T)(T^\dagger X S)(K_SS^\dagger )\). The identities \(\lVert TK_T\rVert =\lVert K_T\rVert ^{1/2}\) and \(\lVert K_SS^\dagger \rVert =\lVert K_S\rVert ^{1/2}\) give the bound. The convergence assertion follows by comparison with \(\sqrt C\lVert T^\dagger X S\rVert \sqrt D\), where \(C,D\) are eventual bounds for the two inverse Gram norms.
For a finite region \(\Lambda \subset \mathbb Z\), include an observable \(X\in \mathcal A_\Lambda \) into a consecutive interval \(I(a,N)\) containing \(\Lambda \) and put
To define the functional before proving independence, one may use \(I(-r,2r+1)\) with \(r=\max _{x\in \Lambda }|x|\) and \(r=0\) when \(\Lambda =\varnothing \). This gives a complex-linear functional without a normalization assumption on \(A\) or \(\rho \).
Suppose \(\sum _i A_i^\dagger A_i=I\) and \(E_A(\rho )=\rho \). Then the preceding definition is independent of the containing interval. For \(\Lambda \subseteq \Gamma \) and \(X\in \mathcal A_\Lambda \), \(\eta _{A,\Gamma }(\iota _{\Lambda ,\Gamma }X)=\eta _{A,\Lambda }(X)\). Neither positivity nor nonzero trace of \(\rho \) is required for these algebraic compatibility identities.
Identity extension on the left preserves expectation by trace preservation; identity extension on the right preserves expectation by \(E_A(\rho )=\rho \). Include two containing intervals in a common larger interval to compare their values. Composition of local inclusions gives finite-region compatibility.
Suppose \(\sum _i A_i^\dagger A_i=I\), \(\rho \geq 0\), \(E_A(\rho )=\rho \), and \(\operatorname {tr}\rho \neq 0\). The functionals \(\eta _{A,\Lambda }\) form a compatible family of normalized positive contractive local states. In particular, \(\eta _{A,I(a,N)}(X)=\eta _A(C_{a,N}X)\) for every \(N\geq 0\). This is the compatible local interpretation of [ 225 , Section 3, equations (3.1)–(3.2b) ] .
The inclusion and coordinate maps preserve identities, products, and adjoints. Positivity and normalization therefore follow from their finite-interval counterparts. Star-algebra homomorphisms between complex C\(^*\)-algebras are contractive, so \(|\eta _{A,\Lambda }(X)|\leq \| X\| \).
Under the preceding hypotheses and \(d{\gt}0\), let \(\omega _A\) be the continuous linear extension of the compatible local MPS state to the completed quasi-local observable algebra.
Under the hypotheses of the preceding definition, the restriction of \(\omega _A\) to \(\mathcal A_\Lambda \) is \(\eta _{A,\Lambda }\). Thus \(\omega _A(X)=\eta _A(C_{a,N}X)\) for an observable on \(I(a,N)\), including \(N=0\). The extension satisfies \(\| \omega _A\| =1\), \(\omega _A(I)=1\), and \(\omega _A(X^*X)\geq 0\) for every quasi-local observable \(X\). No primitivity or purity is asserted.
Apply the completion theorem for compatible local states. Its finite-region evaluation formula gives the asserted restriction, and interval evaluation gives the original trace insertion formula.
For \(a,b\in \mathbb Z\) and \(N\geq 0\), translation by \(b\) sends \(I(a,N)\) to \(I(a+b,N)\). If \(T_b\) denotes the induced local star-algebra equivalence, then \(C_{a+b,N}(T_bX)=C_{a,N}(X)\). The equality includes the empty interval.
The site \(a+i\) is sent to \(a+b+i\) and retains its numbered coordinate \(i\). Hence the two interval coordinate maps give the same configurations and matrix entries. The local inclusion identifying the translated region with \(I(a+b,N)\) has empty complement.
Suppose \(\sum _i A_i^\dagger A_i=I\) and \(E_A(\rho )=\rho \). For every finite region \(\Lambda \subset \mathbb Z\), displacement \(b\in \mathbb Z\), and observable \(X\in \mathcal A_\Lambda \), \(\eta _{A,\Lambda +b}(T_bX)=\eta _{A,\Lambda }(X)\). Positivity and nonzero trace of \(\rho \) are unnecessary for this identity.
Choose an interval containing \(\Lambda \). Translation commutes with the local inclusion and preserves numbered interval coordinates. Evaluate the two local expectations in this interval and its translate.
Suppose \(d{\gt}0\), \(\sum _i A_i^\dagger A_i=I\), \(\rho \geq 0\), \(E_A(\rho )=\rho \), and \(\operatorname {tr}\rho \neq 0\). For every displacement \(b\in \mathbb Z\) and quasi-local observable \(X\), the completed state satisfies \(\omega _A(T_bX)=\omega _A(X)\). This proves translation invariance of the local GVBS state in [ 225 , Section 3, equations (3.1)–(3.2b) ] ; no purity conclusion is asserted.
The equality holds on every finite-region algebra. Translation and the completed expectation are continuous, so it extends from the dense algebraic local algebra to its completion.
Suppose \(d{\gt}0\), \(\sum _i A_i^\dagger A_i=I\), \(\rho \geq 0\), \(E_A(\rho )=\rho \), and \(\operatorname {tr}\rho \neq 0\). For every \(a\in \mathbb Z\), \(k\geq 0\), and \(X\in M_{d^k}(\mathbb C)\), the completed state satisfies \(\omega _A(j_{a,k}(X))=\eta _A(X)\).
Apply interval evaluation after transporting the matrix observable to the interval algebra. The inverse coordinate maps cancel.
Let \(d,D{\gt}0\), and let \(A\) be a normalized primitive tensor with a positive-definite invariant matrix \(\rho \). Suppose that \(h\geq 0\) has kernel \(G_R(A)\) and \(R\geq D^4+1\). There is a constant \(\gamma {\gt}0\) such that, for every \(a\in \mathbb Z\), \(k\geq 1\), and \(X\in M_{d^k}(\mathbb C)\) with \(\omega _A(j_{a,k}(X))=0\),
where \(t=R-1\), \(w=2t+k\), \(B=I_t\otimes X\otimes I_t\), and \(Y=B^*[H_w(h),B]\). The constant is independent of \(a\), \(k\), and \(X\). This is the local commutator inequality in the constructed primitive quasi-local state, at sufficient range. Identification of all pure multiblock GVBS states and construction of the GNS Hamiltonian remain separate from this consequence of [ 225 , Theorem 1.2 and Section 6 ] .
The interval inclusion preserves products and adjoints. Evaluate both sides with the trace insertion formula and apply the finite-interval primitive-sector bound.
Let \(A\) be an MPS tensor, let \(1\leq L\leq N\), and denote its open ground projection on \(n\) sites by \(P_A(n)\). If \(Q_A(L,N)\) is \(P_A(L)\) placed on the first \(L\) sites and extended by the identity, then
No injectivity or normalization is assumed.
Every open MPS vector satisfies the parent constraint on its prefix. Thus the complementary prefix projection annihilates the full ground space. Taking adjoints gives the identity in the other order.
Let \(A,B\) be arbitrary MPS tensors with the same physical dimension. Let \(X\) act on \(k\geq 1\) sites, and let \(\ell \geq 1\) and \(r\geq 0\). Write \(N=\ell +k+r\) and let \(X_{\ell ,r}\) be the placement of \(X\) after the first \(\ell \) sites. Then
The observable commutes with both prefix projections. The preceding identity therefore gives \(P_AX_{\ell ,r}P_B=P_AQ_AQ_BX_{\ell ,r}P_B\). The norm of \(Q_AQ_B\) is bounded by the norm of the prefix product, while the remaining ground projections are contractions.
Let \(A,B\) be trace-preserving primitive MPS tensors with positive definite invariant matrices and positive bond dimensions. Suppose that \(A\) and \(B\) are inequivalent under a bond similarity and a phase, including any identification of equal bond dimensions. Then
For any fixed observable \(X\) on \(k\geq 1\) sites, any filter along which \(\ell \longrightarrow \infty \), and any choice of \(r\geq 0\), one also has
No growth condition is required of the right interval. This is a locality consequence of the overlap estimate in [ 225 , proof of the lemma on distinct local support spaces ] .
The relative overlap estimate bounds the product of the two ground projections by an arbitrary positive number at sufficiently large length. The second assertion follows by multiplying this bound by the fixed norm of the observable and applying the preceding lemma.
Let \(G_{j,n}\) be a finite family of subspaces of finite-dimensional complex Hilbert spaces \(E_n\), along an arbitrary filter. Suppose that for every \(\varepsilon {\gt}0\), eventually every pair \(i\neq j\) satisfies \(|\langle x,y\rangle |\leq \varepsilon \| x\| \| y\| \) for \(x\in G_{i,n}\) and \(y\in G_{j,n}\). Write \(P_{j,n}\) for their orthogonal projections and \(P_n\) for the projection onto their joint span. Then \(\| \sum _jP_{j,n}-P_n\| \to 0\).
Let \(C_{ij}=1\) for \(i\neq j\) and \(C_{ii}=0\). The overlap matrix \(B=\varepsilon C\) satisfies \(\| B\| \to 0\). The finite-family comparison bounds the projection error by \(\delta /(1-\delta )\), where \(\delta =\| B\| /(1-\| B\| )\). This bound tends to zero.
Under the preceding asymptotic overlap hypothesis, let \(W_n\subseteq E_n\) have orthogonal projection \(Q_n\), and let the operators \(X_n\) be eventually uniformly bounded. If \(\| P_{j,n}X_nQ_n\| \to 0\) for every sector \(j\), then \(\| P_nX_nQ_n\| \to 0\). The input space \(W_n\) may be arbitrary and need not be a member of the finite family.
Since \(\| Q_n\| \leq 1\),
Both terms tend to zero, by the projection comparison and finiteness of the sector family.
Let \(A_j\) be finitely many normalized primitive tensors with positive bond dimensions \(D_j\), positive-definite invariant matrices \(\rho _j\), and pairwise gauge-and-phase inequivalence whenever their bond dimensions agree. Fix a sector \(\alpha \) and an observable \(X\) on \(k{\gt}0\) sites with \(\eta _{A_\alpha }(X)=0\). Along an arbitrary filter let \(\ell _n,r_n\to \infty \) and \(N_n=\ell _n+k+r_n\). If \(P_n\) projects onto \(\sum _jG_{N_n}(A_j)\) and \(P_{\alpha ,n}\) onto \(G_{N_n}(A_\alpha )\), then
This supplies the full finite-ground-space projection term in [ 225 , Section 6 ] . No finite-chain uniqueness or purity of a completed state is inferred.
The sector overlaps tend uniformly to zero by inequivalence and primitivity. The diagonal compression tends to zero because the observable is centered in sector \(\alpha \). Every off-diagonal compression tends to zero by prefix locality. Interior placement is contractive, so the finite-sector assembly theorem applies.
Let \(A\) be an MPS tensor and let \(X\) act on \(k\) sites. If \(G_k(A)\subseteq \ker X\), then \(\eta _A(X)=0\) for every virtual matrix \(\rho \), without normalization or positivity assumptions. If \(d{\gt}0\), \(\sum _i A_i^\dagger A_i=I\), \(\rho \geq 0\), \(E_A(\rho )=\rho \), and \(\operatorname {tr}\rho \neq 0\), this gives \(\omega _A(j_{a,k}(X))=0\) for every \(a\in \mathbb Z\). In particular, the constructed state has expectation zero for every translate of a positive local parent interaction. This is the local support property in [ 225 , Theorem 1.1 and Section 3 ] .
The hypothesis gives \(X\Gamma _k=0\), and hence \(\Gamma _k^*X\Gamma _k=0\). The latter operator is the Choi reshuffling of the insertion transfer map. Its matrix-unit coefficients therefore imply that this map is zero. Taking the trace at \(\rho \) proves the first assertion; interval evaluation proves the assertions for the completed state.
Let \((A_j,\rho _j)_{j\in J}\) be a finite family of pairwise inequivalent trace-preserving primitive tensors with positive bond dimensions and positive-definite invariant matrices. Write \(G(N)=\sum _jG_{A_j}(N)\) and let \(P(N)\) be its orthogonal projection. Fix a sector \(\alpha \), an observable \(X\) on \(k\geq 1\) sites with \(\eta _\alpha (X)=0\), and free lengths \(\ell ,r\longrightarrow \infty \). For any varying boundary vectors \(\psi \in G_{A_\alpha }(\ell +k+r)\) with eventually \(\| \psi \| \leq 1\),
The assertion holds along an arbitrary filter.
The sector projection fixes \(\psi \). Thus the vector norm is bounded by the norm of the compression from that sector into the full ground space, multiplied by \(\| \psi \| \). Apply the compression limit and square.
Under the preceding tensor hypotheses, suppose that \(H_n\) are eventually positive operators on chains of length \(N_n=\ell _n+k+r_n\), with \(\ell _n,r_n\longrightarrow \infty \), and eventually
Let \(\psi _n\in G_{A_\alpha }(N_n)\) be eventually unit vectors, and let \(B_n=X_{\ell _n,r_n}\) with \(\eta _\alpha (X)=0\). If
then
The distinguished sector lies in the full ground space. Primitive convergence gives \(\| B_n\psi _n\| ^2\to \operatorname {Re}\eta _\alpha (X^*X)\), and the preceding theorem removes its full ground component. Apply the positive finite-volume gap inequality and pass to the limit.
Under the same finite-family tensor hypotheses, let \(h\) be a positive interaction on \(R\geq 1\) sites. Put \(t=R-1\), \(N_n=(n+t+k)+(t+n)\), and let \(H_n\) be its open interaction sum. Suppose that these operators eventually have kernel \(G(N_n)\) and norm gap at least \(\gamma \geq 0\). For eventually unit \(\psi _n\in G_{A_\alpha }(N_n)\) and every \(k\geq 1\) observable \(X\) with \(\eta _\alpha (X)=0\),
Finite-range locality places the commutator energy on the fixed interval of length \(t+k+t\). Primitive expectation convergence determines its limit, so the preceding reduction applies without a supplied energy limit.
Let \(A_j\) be trace-preserving primitive tensors of positive bond dimensions, with positive definite invariant matrices \(\rho _j\), and let \(T=\bigoplus _j\mu _jA_j\), where every \(\mu _j\neq 0\). Assume that the simultaneous word span is the full product of the bond matrix algebras at a length \(S\geq 1\). Let \(R\geq S+1\), and let \(h\) be a positive interaction on \(R\) sites with \(\ker h=G_T(R)\). There exists \(\gamma {\gt}0\) such that, for every sector \(\alpha \) and every observable \(X\) on any \(k\geq 1\) sites with \(\eta _\alpha (X)=0\),
In particular this holds for the canonical parent projection \(h=I-P_T(R)\). These are finite-interval sector expectation statements; they do not assert the construction of a GNS Hamiltonian or purity of its state. The range is sufficient and need not be optimal. [ 225 , Theorem 1.2 and Section 6 ] , [ 61 ] .
Simultaneous injectivity implies sector inequivalence and identifies the open-chain kernel with the sum of the sector spaces at this range. The open-chain theorem supplies a uniform canonical gap; positivity and the prescribed local kernel transfer it to \(h\). Choose eventual unit boundary vectors independently in each sector and apply the preceding local inequality. The same gap constant works for every sector and every observable.
Let \(d{\gt}0\), and let \(A_j\) be finitely many normalized primitive tensors with positive bond dimensions and positive-definite invariant matrices \(\rho _j\). Suppose that the simultaneous word span is full at a length \(S{\gt}0\). Let \(T=\bigoplus _j\mu _jA_j\) with \(\mu _j\neq 0\), and let \(h\geq 0\) have kernel \(G_R(T)\) at range \(R\geq S+1\). There is a constant \(\gamma {\gt}0\) such that, for every sector \(\alpha \), \(a\in \mathbb Z\), \(k\geq 1\), and \(X\in M_{d^k}(\mathbb C)\) with \(\omega _{A_\alpha }(j_{a,k}(X))=0\),
where \(t=R-1\), \(w=2t+k\), \(B=I_t\otimes X\otimes I_t\), and \(Y=B^*[H_w(h),B]\). The same constant applies to all sectors, positions, and centered observables. The states are the completed expectations of the given primitive blocks; identification of the full pure GVBS ground-state face remains a separate assertion.
The common finite-interval bound is expressed in the constructed states by interval evaluation. The interval inclusions preserve products and adjoints, so the centered hypothesis and both terms agree exactly.
For tensors \(A\) and \(B\) with a common physical alphabet and possibly different bond dimensions, define the insertion of a \(k\)-site matrix \(X\) by
Here \(Y\) is a rectangular matrix from the virtual space of \(A\) to that of \(B\). For \(A=B\), this is the ordinary observable insertion.
The coefficient of \(\Gamma _{A,k}^*X\Gamma _{B,k}\) between boundary units \((b,a)\) and \((e,c)\) is \(E_{B,A;X}(|c\rangle \langle a|)_{eb}\). Moreover,
and
Expand the boundary traces in matrix units. Splitting a physical word at a prescribed cut gives the composition identity, and identity insertions give the two ordinary transfer powers.
Suppose the rectangular mixed transfer map \(E_{B,A}\) has spectral radius less than one. Then \(E_{B,A}^{\, n}\to 0\) in operator norm. For a fixed local matrix \(X\), and along any filter with \(\ell \to \infty \) and \(r\to \infty \),
in operator norm.
The mixed transfer powers decay pointwise by the strict spectral-radius bound. Finite dimension upgrades this to operator norm convergence; continuity of composition then gives the inserted sandwich limit.
Under the preceding strict mixed spectral-radius hypothesis, let \(N=\ell +k+r\) and \(X_{\ell ,r}=1_\ell \otimes X\otimes 1_r\). As \(\ell ,r\to \infty \) along any common filter,
No injectivity of the boundary maps is required. This is a distinct-sector estimate, rather than an assertion of the infinite-volume spectral gap.
The pullback is the rectangular coefficient reshuffling of the virtual sandwich. This reshuffling is linear between fixed finite-dimensional spaces and hence continuous. Apply it to the sandwich limit.
Let \(d{\gt}0\), and let \(A_j\) have positive bond dimensions and full simultaneous word span at \(S{\gt}0\). Let \(T=\bigoplus _j\mu _jA_j\) with \(\mu _j\neq 0\), and let \(h\geq 0\) have kernel \(G_R(T)\) at \(R\geq S+1\). There exist normalized primitive representatives \(B_j\) with positive-definite invariant matrices \(\rho _j\) such that \(G_N(A_j)=G_N(B_j)\) for every \(j,N\). Their constructed quasi-local states satisfy the preceding common commutator inequality for the original interaction \(h\), with one positive constant for all sectors, interval positions, and centered local observables. No normalized representatives or invariant matrices are supplied.
Primitive normalization preserves each block support and simultaneous injectivity. Hence it also preserves the assembled local support and the parent-interaction kernel. Apply the common bound to the normalized family with the original positive interaction.
Let \(j_{a,k}\) denote inclusion of the interval \([a,a+k)\) into the quasi-local algebra. Then, including empty intervals,
For \(k{\gt}0\), the nonwrapping inclusion of \(X\) at position \(b\) in a chain of length \(N\), where \(b+k\leq N\), has quasi-local image \(j_{a+b,k}(X)\) when the chain begins at \(a\).
Interval coordinates identify identity extensions with finite-region inclusions. The quasi-local inclusions agree under enlargement of the finite region. The nonwrapping statement is obtained by adjoining the remaining sites on either side.
Let \(R,k{\gt}0\), \(t=R-1\), \(w=k+2t\), and let \(h\) be an arbitrary \(R\)-site matrix. Write \(H_N\) for its open-chain sum and \(X_N=I_\ell \otimes X\otimes I_r\), with \(\ell ,r\geq t\) and \(N=\ell +k+r\). Let
Then
Equivalently, putting \(x=j_{a,k}(X)\) and \(\widehat H_N=j_{a-\ell ,N}(H_N)\) gives \(x^*[\widehat H_N,x]=j_{a-t,w}(Y)\). Neither positivity nor a state assumption is required.
The finite-chain energy is a nonwrapping inclusion of the fixed patch observable \(Y\). Its quasi-local image is therefore \(j_{a-t,w}(Y)\). The equivalent form follows because the inclusions preserve products and adjoints.
Under the preceding assumptions, if \(\ell ,r\to \infty \) along any common filter, the quasi-local energy is eventually equal to \(j_{a-t,w}(Y)\) and hence converges to it in norm. Consequently every functional \(\omega \) on the quasi-local algebra satisfies
Continuity is unnecessary here because the observable is eventually constant.
Both growing margins eventually exceed \(t\). Apply the fixed-support identity on that tail; the convergence statements follow from eventual constancy.
For an MPS tensor \(A\), let \(p_A(N)\) be the matrix of the orthogonal projection onto \(G_A(N)\) in numbered coordinates on \(N\) sites. It is a self-adjoint idempotent. Its inclusion \(j_{a,N}(p_A(N))\) into the quasi-local algebra on the interval beginning at \(a\) is also a self-adjoint idempotent, including when \(N=0\).
Coordinate transport and the interval inclusion preserve products and adjoints, hence preserve orthogonal projections.
Let \(f\) be a positive complex-linear functional on a star-ordered complex algebra in which every element admits a decomposition into self-adjoint real and imaginary parts. If \(p=p^*=p^2\) and \(f(p)=f(1)\), then, for every \(X\),
In particular, let \(\varphi \) be a normalized positive norm-one state on the quasi-local algebra. For an interval projection \(p\) with \(\varphi (j_{a,N}(p))=1\), an interval observable \(X\), and \(c\in \mathbb C\),
No translation invariance is assumed.
Put \(q=1-p\). Then \(f(q)=f(q^*q)=0\). Cauchy–Schwarz gives \(f(qX)=f(Xq)=0\), which proves the compression identity. Restrict the quasi-local state to the numbered interval matrix algebra, apply this identity, and use its norm-one bound on \(pXp-cp\). The interval inclusion preserves the operator norm.
Let \(A\) be a trace-preserving primitive MPS tensor of positive physical and bond dimensions, with positive definite invariant matrix \(\rho \). Let \(\omega _A\) be its constructed quasi-local state. Suppose that \(\varphi \) is any normalized positive norm-one continuous complex-linear state on the quasi-local algebra and that
Then every interval observable has the prescribed insertion expectation:
and consequently \(\varphi =\omega _A\). The competing state need not be translation invariant. This uniqueness statement is a consequence of primitive boundary convergence, as used in [ 225 , Sections 3 and 6 ] ; purity is a separate consequence of uniqueness and inheritance of support in convex decompositions.
Include a fixed interval observable into intervals expanded by \(n\) sites on either side. Write \(P_n\) for the ground projection of the expanded interval and \(X_n\) for the included observable. The preceding estimate gives
Primitive scalar compression makes the right-hand side tend to zero. Thus the states agree on every interval observable. Every finite-region observable embeds into a containing interval, and their union is dense in the quasi-local algebra. Continuity proves equality on the completion.
Let \(d{\gt}0\) and let \(A\) be trace preserving. Let \(\rho \ge 0\) be an invariant virtual matrix of nonzero trace. For every \(a\in \mathbb {Z}\) and \(N\ge 0\), the constructed state satisfies \(\omega _A(j_{a,N}(P_{G_N(A)}))=1\). Primitivity and faithfulness of \(\rho \) are not required.
The matrix \(I-P_{G_N(A)}\) annihilates \(G_N(A)\), so its translated expectation is zero by the local support theorem. Since \(\omega _A(I)=1\), linearity gives expectation one on the projection.
Let \(d,D{\gt}0\), let \(A\) be a trace-preserving primitive tensor, and let \(\rho {\gt}0\) be its invariant virtual matrix. The completed MPS state \(\omega _A\) is pure among all states of the quasi-local algebra. The states in a proposed convex decomposition need not be translation invariant. This proves the primitive canonical case of the purity assertion in [ 225 , Section 3 ] ; no classification of the multiblock ground-state face is asserted here.
Apply the supported-state extremality criterion to the family of all translated finite-interval MPS projections. They have expectation one in \(\omega _A\), and every state satisfying the same support constraints equals \(\omega _A\) by primitive supported-state uniqueness.
Let \(A\) be a tensor, let \(\rho \geq 0\) be a virtual matrix, and let \(H\geq 0\) act on \(N\) physical sites with \(G_N(A)\subseteq \ker H\). For every \(N\)-site matrix \(B\),
In particular, the commutator expectation is real. Neither primitivity nor a spectral gap is needed.
The matrix \(B^*BH\) annihilates \(G_N(A)\), so its expectation vanishes. The remaining matrix \(B^*HB\) is positive, and the insertion expectation is a positive linear functional.
Let \(T\) be the weighted sum of tensors \(A_\alpha \), with nonzero weights. Suppose \(h\geq 0\) acts on \(R{\gt}0\) sites and has kernel \(G_R(T)\). For a \(k\)-site matrix \(X\) with \(k{\gt}0\), put \(t=R-1\), \(w=k+2t\), and
where \(H_w\) is the open-chain sum of \(h\). For every block \(\alpha \) and every virtual matrix \(\rho _\alpha \geq 0\),
If \(A_\alpha \) is trace preserving and \(\rho _\alpha \) is an invariant matrix of nonzero trace, the corresponding constructed quasi-local state satisfies \(\omega _\alpha (j_{a,w}(Y))\geq 0\) for every interval position \(a\). No simultaneous injectivity bound is imposed.
The positive parent sum and the canonical open Hamiltonian have the same kernel. The canonical open Hamiltonian annihilates \(G_w(T)\), which contains \(G_w(A_\alpha )\). Apply the preceding lemma to \(H_w\) and the padded observable. The quasi-local statement follows from exact local evaluation of the constructed state.
Let \(A=\bigoplus _{\alpha =0}^{b-1}\mu _\alpha A_\alpha \), with nonzero coefficients, trace-preserving primitive blocks, and faithful invariant density matrices. Suppose that the blocks have simultaneous injectivity length \(S{\gt}0\), and that \(h\geq 0\) is a parent interaction of range \(R\geq S+1\). Then there is \(\gamma {\gt}0\) such that, for every constructed sector state \(\omega _\alpha \) and every local observable \(X\) satisfying \(\omega _\alpha (X)=0\),
Here \(X\) is supported on \([a,a+k)\), with \(k{\gt}0\), and the limit may be taken along any filter for which both margins tend to infinity. The limit is real and nonnegative. This proves the displayed infinite-volume estimate of [ 225 , Theorem 1.2 ] for these constructed sectors at the stated sufficient range; identification of the entire source ground-state face is a separate result.
Once both margins contain \(R-1\) sites, the quasi-local energy observable equals the inclusion of the fixed commutator patch. Thus its expectation is eventually constant. Local support and positivity make this expectation real and nonnegative. The common sector gap bounds its real part by \(\gamma \omega _\alpha (X^*X)\), and positivity of the state makes the latter real as well.
Let \(J\) be finite, let \(G_j(n)\) be subspaces of finite-dimensional complex Hilbert spaces, and let \(P_j(n)\) and \(P(n)\) be the projections onto \(G_j(n)\) and \(\sum _jG_j(n)\), respectively. Fix \(c_j\in \mathbb {C}\). Suppose \(\| \sum _jP_j(n)-P(n)\| \to 0\), the operators \(X_n\) are eventually uniformly bounded, and \(\| P(n)X_nP_j(n)-c_jP_j(n)\| \to 0\) for every \(j\). Then \(\| P(n)X_nP(n)-\sum _jc_jP_j(n)\| \to 0\) along the same filter.
Write \(Q=\sum _jP_j\). Expanding the rightmost joint projection gives \(PXP-\sum _jc_jP_j=PX(P-Q)+\sum _j(PXP_j-c_jP_j)\). Since \(\| P\| \le 1\),
The finite sum tends to zero, as does the projection error multiplied by the uniform norm bound. Apply (425).
Let \((A_j)_{j\in J}\) be a finite family of trace-preserving primitive tensors of positive bond dimensions, with faithful invariant matrices \(\rho _j\). Suppose distinct tensors are inequivalent up to virtual gauge and phase. Fix a sector \(\alpha \), an observable \(X\) on \(k{\gt}0\) sites, and margins \(\ell _n,r_n\to \infty \) along any filter. Put \(N_n=\ell _n+k+r_n\), let \(P_j(n)=P_{G_{N_n}(A_j)}\), let \(P(n)\) project onto \(\sum _jG_{N_n}(A_j)\), and put \(X_n=I_{\ell _n}\otimes X\otimes I_{r_n}\). Then \(\| P(n)X_nP_\alpha (n)-\eta _{A_\alpha }(X)P_\alpha (n)\| \to 0\).
Center \(X\) as \(Y=X-\eta _{A_\alpha }(X)I\). Linearity and normalization give \(\eta _{A_\alpha }(Y)=0\). Interior inclusion preserves this centering, and \(P(n)P_\alpha (n)=P_\alpha (n)\). The centered full compression limit therefore gives the claimed limit.
Under the hypotheses and notation of Theorem 12.9.84, with no distinguished sector, one has \(\| P(n)X_nP(n)-\sum _j\eta _{A_j}(X)P_j(n)\| \to 0\). The finite-length projections need not be mutually orthogonal.
The sector overlap limits and \(N_n\to \infty \) imply \(\| \sum _jP_j(n)-P(n)\| \to 0\). Interior inclusion gives \(\| X_n\| \le \| X\| \). Apply the joint-compression theorem with \(c_j=\eta _{A_j}(X)\) and the sector column limits.
Let \(f\) be a positive normalized complex-linear functional and \(p=p^*=p^2\). If \(f(H)=0\) and \(\kappa (1-p)\leq H\) for some \(\kappa {\gt}0\), then \(f(p)=1\). In particular, if \(H\geq 0\) is a finite-interval Hamiltonian with \(\ker H=G_N(A)\), then every normalized positive quasi-local state satisfying \(\varphi (j_{a,N}(H))=0\) also satisfies \(\varphi (j_{a,N}(p_A(N)))=1\). No uniform gap is required. This is the finite-dimensional support implication used in [ 225 , Section 3 ] .
Positivity gives \(f(1-p)\geq 0\), whereas monotonicity gives \(\kappa f(1-p)\leq f(H)=0\). Thus \(f(1-p)=0\) and \(f(p)=1\). A finite positive Hamiltonian dominates a positive multiple of its kernel-complement projection. Restrict the quasi-local state to the interval matrix algebra and apply the preceding argument.
Let \(A=\bigoplus _\alpha \mu _\alpha A_\alpha \) have nonzero weights and simultaneous injectivity length \(S{\gt}0\). Let \(h\geq 0\) have \(\ker h=G_R(A)\) at range \(R\geq S+1\). If a normalized positive quasi-local state \(\varphi \) satisfies \(\varphi (j_{a,R}(h))=0\) for every \(a\in \mathbb Z\), then
Consequently the same support constraint holds eventually on the symmetric intervals \([-n,n+1)\). The state need not be translation invariant, and primitive normalization of the original blocks is unnecessary. This is the local support consequence of the zero-energy condition in [ 225 , Theorem 1.1 ] , at the stated sufficient range.
Every finite open sum has zero expectation, since each translated local term does. Positivity is preserved by the sum, and its kernel is exactly \(G_N(A)\) when \(N\geq R\). The preceding support lemma applies. Since \(2n+1\geq R\) eventually, it also applies to the symmetric intervals.
Let \(h\) be an \(R\)-site matrix, where \(R{\gt}0\), and let \(H_N\) be its open-chain sum. For every interval position \(a\) and every \(N\geq 0\),
with the sum understood as empty when \(N{\lt}R\). Consequently, any complex-linear functional \(\varphi \) satisfying \(\varphi (j_{a,R}(h))=0\) for every \(a\in \mathbb Z\) also satisfies
for every \(a\) and \(N\). This applies in particular to arbitrary continuous states, without requiring translation invariance.
Every nonwrapping interaction window has quasi-local image \(j_{a+i,R}(h)\). The interval inclusion preserves the finite sum. Apply linearity of \(\varphi \) and the vanishing hypothesis to its summands.
Let \(\varphi \) be a continuous complex-linear functional on the quasi-local algebra that is nonnegative on every element \(Y^*Y\). Its restriction to the numbered matrix algebra on an interval is the positive functional \(f_{a,N}(X)=\varphi (j_{a,N}(X))\).
Every positive matrix is a square \(Y^*Y\). The interval inclusion preserves products and adjoints, so the restriction is positive.
Suppose \(\varphi \) is a normalized positive norm-one quasi-local state. For an interval projection \(p=p^*=p^2\), its expectation \(\varphi (j_{a,N}(p))\) is real and belongs to \([0,1]\). Every interval observable satisfies \(|\varphi (j_{a,N}(X))|\leq \| X\| \). If \(\varphi (j_{a,N}(p))=1\), then
No translation invariance is assumed.
Positivity of \(p\) and \(1-p\) gives the interval bounds and realness. The norm-one condition and the isometric interval inclusion give the operator-norm estimate. The support-compression identity reduces the final difference to the expectation of \(pXp-Y\).
Let \(J\) be finite, and let \(w_j(n)\) be real weights that eventually lie in \([0,1]\), with \(\sum _jw_j(n)\to 1\). There are a strictly increasing subsequence \(n_s\) and weights \(p_j\geq 0\), \(\sum _jp_j=1\), such that \(w_j(n_s)\to p_j\) for every \(j\). For any family of tests \(x\), if \(f(x)-\sum _jw_j(n)g_j(x)\longrightarrow 0\) for every test \(x\), then one choice of normalized weights satisfies \(f(x)=\sum _jp_jg_j(x)\) for all tests. The test family need not be countable.
Extract a convergent subsequence in the compact cube \([0,1]^J\). Continuity of the finite sum shows that the limit has total mass one. For each test, compare the assumed limit with the evaluation along this same subsequence. Uniqueness of limits gives the asserted identity.
Let the original block family have simultaneous injectivity length \(S{\gt}0\) and nonzero weights. For every positive parent interaction of the original weighted tensor at range \(R\geq S+1\), there are trace-preserving primitive representatives, with faithful invariant matrices, preserving every block ground space at every length. Their completed sector states are pure among all quasi-local states and have a common positive literal infinite-volume commutator gap for the original interaction. The original blocks need not be normalized, and the invariant matrices and purity are conclusions. The two free margins may tend to infinity along any filter, as in the preceding theorem.
Normalize the blocks without changing their local ground spaces or their simultaneous span. Equality of the block spaces preserves the local kernel of the original positive interaction. Apply the common sector commutator theorem to the representatives and the primitive purity theorem to each completed sector state.
Let \(h\geq 0\) have positive finite interaction range. Suppose that \(\ker H_N(h)=G_N(A)\) for every sufficiently large \(N\). If a normalized positive quasi-local state \(\varphi \) has \(\varphi (j_{a,R}(h))=0\) for every \(a\in \mathbb Z\), then, for every sufficiently large \(n\),
This uses the eventual kernel hypothesis in [ 225 , Theorem 1.2 ] directly. It imposes no local parent-kernel equality and no lower bound on the interaction range in terms of injectivity.
Every open interaction sum has zero expectation. For sufficiently large symmetric intervals it is positive and has the prescribed MPS kernel. Its finite-dimensional kernel-complement comparison therefore forces expectation one on the MPS support projection.
Let \(d{\gt}0\) and let \(h\) be an observable on \(R\) consecutive sites. Define \(\mathcal F_h\) to be the set of all states \(\phi \) of the quasi-local algebra such that \(\phi (j_{a,R}(h))=0\) for every \(a\in \mathbb {Z}\). Translation invariance of \(\phi \) is not required. For a positive parent interaction, this is its zero-energy state set, as in [ 225 , Section 1 and Theorem 1.1 ] .
The set \(\mathcal F_h\) is real convex for every local observable \(h\). If \(h\ge 0\), it is a face of the full state space: whenever \(\omega =(1-t)\phi +t\psi \in \mathcal F_h\), with \(0{\lt}t{\lt}1\) and \(\phi ,\psi \) states, both constituents belong to \(\mathcal F_h\).
The zero-expectation equations are linear, and the state space is convex. For the face assertion factor \(h=Y^*Y\). Each translate is the adjoint square \(j_{a,R}(Y)^*j_{a,R}(Y)\). Vanishing of its expectation in a nontrivial convex combination forces vanishing in each constituent, by positivity.
Let \(A\) be trace preserving and let \(\rho \ge 0\) be invariant with nonzero trace. If \(G_R(A)\subseteq \ker h\), then the constructed state \(\omega _A\) belongs to \(\mathcal F_h\). Consequently, if \(B=\bigoplus _j\mu _jA_j\), with all \(\mu _j\ne 0\), and \(h\) is a positive parent interaction for \(B\), every sector for which such trace-preserving and invariant-matrix data are given determines a state in \(\mathcal F_h\). Neither primitivity nor faithfulness of the invariant matrix is required.
Exact local support gives \(\omega _A(j_{a,R}(h))=0\) at every position, while normalization and positivity make \(\omega _A\) a state. For the weighted block tensor, \(G_R(A_j)\subseteq \sum _iG_R(A_i)=G_R(B)=\ker h\). Apply the same support assertion to each sector.
Let \((A_j)_{j=0}^{b-1}\) be finitely many trace-preserving primitive tensors of positive physical and bond dimensions, with positive definite invariant matrices \(\rho _j\). Suppose distinct sectors are inequivalent up to virtual gauge and phase. Let \(T\) be their weighted block sum with nonzero coefficients, and let \(\omega _j\) be the constructed quasi-local state of the \(j\)th sector. Suppose \(\varphi \) is a normalized positive norm-one continuous complex-linear state on the quasi-local algebra and that, for all sufficiently large \(n\),
Then there are real weights \(w_j\geq 0\), \(\sum _jw_j=1\), such that
The competing state need not be translation invariant. This is the support-to-convex-hull implication in the argument for [ 225 , Theorem 1.1 ] .
On \(I_n=[-n,n+1)\), put \(w_j(n)=\varphi (j_{-n,2n+1}(p_{A_j}(2n+1)))\). Positivity gives real weights in \([0,1]\). The sum of the sector projections approaches the joint projection in operator norm. The support condition and the norm-one bound therefore give \(\sum _jw_j(n)\to 1\).
Fix an observable \(X\) on \(I_m\). For \(n\geq m\), include it into \(I_n\) with equal margins \(n-m\). Support compression and asymptotic sector decomposition give
Extract one convergent subsequence of the finite weight vectors. Its limit is nonnegative with total mass one. The same subsequence identifies all centered interval expectations with the corresponding convex combination. Every finite region lies in such an interval, and local observables are dense. Continuity proves the state identity on the completed algebra.
Let \(T=\bigoplus _j\mu _jA_j\) with every \(\mu _j\neq 0\), and let \(A_\alpha \) be a trace-preserving block with positive invariant density \(\rho _\alpha \) of nonzero trace, and let \(\omega _\alpha \) be its constructed quasi-local state. Let \(h\) act on \(R{\gt}0\) sites. Suppose that, for some \(N\geq R\), the joint MPS space \(G_N(T)\) is contained in \(\ker H_N\), where \(H_N\) is the open-chain sum of \(h\). Then \(\omega _\alpha (j_{a,R}(h))=0\) for every interval position \(a\). Positivity of \(h\) is unnecessary.
The sector space is contained in the joint MPS space, so \(\omega _\alpha (j_{a,N}(H_N))=0\). Exact interval evaluation gives the same expectation \(\eta _\alpha (h)\) for each translate of \(h\). Thus \((N-R+1)\eta _\alpha (h)=0\). Since \(N\geq R\), the coefficient is positive.
Let \(T=\bigoplus _j\mu _jA_j\) with every \(\mu _j\neq 0\). For every block \(\alpha \) and every length \(L\), the boundary space \(G_L(A_\alpha )\) is contained in \(G_L(T)\) [ 225 , Lemma commutation (ii) ] .
With nonzero weights, the joint boundary space is the sum of the block boundary spaces.
Under the same sector-state assumptions, suppose that \(\ker H_N=\mathcal G_N\) for every sufficiently large \(N\), where \(\mathcal G_N\) is the joint open-boundary MPS space. Then \(\omega _\alpha (j_{a,R}(h))=0\) for every \(a\) and every sector \(\alpha \). No injectivity condition at the interaction range is required. This is the sector-membership consequence of the finite-volume kernel hypothesis in [ 225 , Theorem 1.2 ] .
Choose a volume \(N\geq R\) at which the kernel equality holds, and apply the finite-volume statement.
Suppose \(W\leq N\), fix a cyclic start \(s\), and let \(q{\lt}W\) satisfy \(q+R\leq W\). Replacing the range-\(R\) subwindow beginning at \(s+q\) changes only the active configuration. Restriction to this subwindow agrees with restriction at \(q\) inside the active block. When \(R{\gt}0\), any local operator, extended by the identity, satisfies
Its action is obtained by applying \(h\) to the restricted window with outside coordinates fixed. If \(0{\lt}R\leq W\leq n\leq N\), translation \(q\mapsto n-W+q\) also identifies the nonwrapping starts on \(W\) sites with the range-\(R\) starts contained in \([n-W,n)\).
In active-block coordinates, subwindow replacement changes only the first coordinate. The start bijection has inverse subtraction of \(n-W\). Applying the local operator separately on each complementary configuration gives the conjugation identity.
For a range-\(R\) operator \(h\), let \(H_{[n-W,n)}(h)\) be the sum of its nonwrapping translates in an \(N\)-site chain whose starts satisfy \(n-W\leq i\) and \(i+R\leq n\).
Let \(R\leq W\). For any positive operators \(T_i\), indexed by the nonwrapping range-\(R\) starts in an \(N\)-site chain,
In particular, for a positive interaction \(h\), \(\sum _{n=W}^{N}H_{[n-W,n)}(h) \leq (W-R+1)H_N^{\mathrm{open}}(h)\). This is the counting argument of condition C1 in [ 225 ] , applied to arbitrary positive terms.
A term beginning at \(i\) can occur only for \(i+R\leq n\leq i+W\), an interval containing \(W-R+1\) integers. Reverse the order of summation and use positivity of each term.
Suppose \(0{\lt}R\leq W\leq n\leq N\). The suffix Hamiltonian is unitarily equivalent to \(H_W^{\mathrm{open}}(h)\) acting independently on the complementary configurations. If \(\kappa p_W(A)\leq H_W^{\mathrm{open}}(h)\), then the same inequality holds after embedding both sides on the suffix \([n-W,n)\). Here \(p_W(A)\) is the canonical complementary ground-space projection; no sign condition on \(\kappa \) or positivity condition on \(h\) is needed for this transport of order.
Sum the conjugation identity over the corresponding start bijection. Extension by the identity and conjugation by an isometry preserve order.
Let \(0{\lt}R\leq W\leq N\), let \(h\) be positive, and suppose \(\kappa p_W(A)\leq H_W^{\mathrm{open}}(h)\). Then
If additionally \(\kappa ,\gamma {\gt}0\), both global kernels equal \(G_N(A)\), and \(\gamma \lVert v\rVert \leq \lVert H_{N,W}^{\mathrm{open}}(A)v\rVert \) for \(v\perp G_N(A)\), then
No local parent-kernel equality is assumed for \(h\). This is a finite-window consequence of condition C1 and the comparison argument in Section 6 of [ 225 ] .
Sum the embedded local comparisons. Each full range-\(W\) parent term occurs once, whereas a range-\(R\) term occurs at most \(W-R+1\) times. For the gap, compare the positive operators \(\kappa H_{N,W}^{\mathrm{open}}(A)\) and \((W-R+1)H_N^{\mathrm{open}}(h)\), whose kernels agree, and divide by the positive window multiplicity.
Let \(A_j\) be a finite family of normalized primitive MPS tensors with faithful fixed density matrices, pairwise inequivalent under gauge and phase, and let \(B=\bigoplus _j\mu _jA_j\), where every \(\mu _j\ne 0\). Let \(h\geq 0\) be an interaction of range \(R\geq 1\). Suppose that \(\ker H^{\mathrm{open}}_{h,N}=\mathcal G_N(B)\) for all sufficiently large \(N\). Then there exists \(\gamma {\gt}0\) such that
for every \(N\in \mathbb N\) and every \(v\perp \ker H^{\mathrm{open}}_{h,N}\). No condition on the local kernel of \(h\), or lower bound on \(R\) beyond \(R\geq 1\), is required. This is the finite-chain conclusion of [ 225 , Theorem 1.2 ] , under its eventual finite-interval kernel hypothesis, for the primitive-sector representation; the passage from more general periodic presentations is separate [ 61 ] .
Choose a canonical interaction range \(W=2p\) above the original range and both kernel thresholds, with canonical gap \(\delta {\gt}0\) uniform in the volume. Finite dimensionality and positivity give \(\kappa P_{\mathcal G_W(B)^\perp }\leq H^{\mathrm{open}}_{h,W}\) for some \(\kappa {\gt}0\). For \(N\geq W\), the finite-window comparison gives
The kernels coincide for these volumes, so the transferred gap is \(\kappa \delta /(W-R+1)\). Taking a minimum with the positive gaps on the kernel complements at the finitely many volumes \(N{\lt}W\) gives one constant for every volume.
Let \(T=\bigoplus _{j=0}^{b-1}\mu _jA_j\), with \(\mu _j\ne 0\), be a finite weighted sum of inequivalent trace-preserving primitive tensors with faithful invariant density matrices. Write \(\omega _j\) for their constructed quasi-local states. Let \(h\geq 0\) have range \(R\geq 1\), and suppose \(\ker H_N^{\mathrm{open}}(h)=G_N(T)\) for all sufficiently large \(N\). Then, for every normalized positive state \(\varphi \),
No translation invariance of \(\varphi \) is assumed. This is the ground-state classification associated with [ 225 , Theorems 1.1–1.2 ] for the stated primitive sector family. More general periodic presentations remain separate [ 61 ] .
Zero local energy and the eventual kernel identity give expectation one on the joint support projections of sufficiently large intervals. The supported-state decomposition gives the convex coefficients. Conversely, each sector state has zero local interaction expectation: its expectation of a sufficiently long open sum is zero, and every summand has the same expectation. Convexity preserves this property.
Under the preceding hypotheses, a state \(\varphi \) is a pure element of \(\mathcal F_h\) if and only if \(\varphi =\omega _j\) for some \(j\). Purity is extremality among all states of the quasi-local algebra.
A pure state in a finite convex combination equals one of the constituent states. Each constructed primitive sector state is pure and has zero local energy.
Let \(h\geq 0\) have range \(R{\gt}0\), and suppose that the kernels of its sufficiently long open-chain Hamiltonians equal the joint boundary spaces of a weighted tensor family with nonzero weights. Let \(\omega _\alpha \) be a constructed sector state, with a trace-preserving tensor and a positive invariant matrix of nonzero trace. For \(X\) on \(k{\gt}0\) sites,
Neither primitivity nor faithfulness of the invariant matrix is needed for this assertion. The local commutator patch has real expectation.
Choose one sufficiently large volume containing both interaction margins. Its Hamiltonian annihilates the sector boundary space and is positive. The expectation of \(B^*[H,B]\) therefore equals that of \(B^*HB\) and is nonnegative. The quasi-local commutator identity identifies this expectation with the fixed local patch.
Let \(A_1,\ldots ,A_b\) be pairwise gauge-phase inequivalent normalized primitive tensors with faithful invariant matrices, and let all weights be nonzero. Let \(h\geq 0\) have arbitrary range \(R{\gt}0\). Suppose that \(\ker H_N^{\mathrm{open}}(h)\) equals the joint boundary-condition space for all sufficiently large \(N\). Then there is \(\gamma {\gt}0\) such that, for every \(N\) and every \(v\perp \ker H_N^{\mathrm{open}}(h)\),
and, for every sector, position, and centered observable \(X\) on \(k{\gt}0\) sites,
The same inequality holds for the finite-interval insertion expectations. No local parent-kernel equality or lower bound on \(R\) in terms of an injectivity length is assumed. This is the primitive-sector specialization of the argument in [ 225 , Theorem 1.2 and Section 6 ] ; the more general GVBS presentations are recorded in [ 61 ] .
Choose the uniform finite-chain constant. Pull the eventual kernel equality along growing buffered volumes. Apply the full joint-ground projection decay and the finite-volume commutator estimate in each sector. Evaluation on consecutive intervals is the insertion expectation.
Under the primitive-family hypotheses of Theorem 12.9.110, choose the same \(\gamma {\gt}0\) for the finite-chain bound. Let \(\ell ,r\) tend to infinity along any filter. For every constructed sector state and centered local \(X\) on \(k{\gt}0\) sites, the literal energies
converge to a real number \(e\) satisfying \(\gamma \omega _\alpha (X^*X)\leq e\). Thus one constant gives both the all-volume finite-chain norm gap and the infinite-volume sector bound. This states the primitive-sector case explicitly; classification of all pure zero-energy states is a separate assertion.
Once both margins exceed \(R-1\), locality makes the commutator observable equal to its fixed patch. Its expectation is real and nonnegative by eventual support. Combine this literal limit with the local bound using the same finite-chain constant.
Let \(A_j\) be a finite family with positive bond dimensions whose word tuples at some length \(S{\gt}0\) span the full product matrix algebra. This is the simultaneous spanning condition of [ 17 , Section IV.C ] . Let \(T_A=\bigoplus _j\mu _jA_j\), with every \(\mu _j\ne 0\). Suppose that \(h\geq 0\) has range \(R\geq 1\) and that \(\ker H^{\mathrm{open}}_{h,N}=G_N(T_A)\) for all sufficiently large \(N\). There exist primitive trace-preserving tensors \(B_j\) with faithful invariant matrices \(\rho _j\) such that their word tuples still span the full product algebra at length \(S\). Write \(T_B=\bigoplus _j\mu _jB_j\). Then
These equalities hold for every block \(j\) and every length \(N\). Let \(\omega _j\) be the quasi-local state constructed from \(B_j,\rho _j\). For every continuous complex-linear functional \(\varphi \) on the quasi-local algebra,
The original interaction range need not exceed \(S\). This combines the normalization in [ 225 , equations (3.1)–(3.2b) ] with the ground-state classification of [ 225 , Theorems 1.1–1.2 ] , for the original simultaneously spanning family. The passage from more general periodic presentations is separate [ 61 ] .
Normalize each block by a nonzero rescaling and a virtual gauge. These operations preserve every block ground space and the simultaneous word span. The weighted joint ground spaces therefore coincide at every length, and the eventual kernel identity transfers to \(T_B\). Simultaneous spanning excludes gauge-phase equivalence between distinct normalized blocks. Apply the convex-combination and pure-sector classifications to this normalized family and the unchanged interaction.
Let \(T=\bigoplus _j\mu _jA_j\) be a finite weighted family of inequivalent trace-preserving primitive tensors with faithful invariant density matrices and nonzero coefficients. Let \(h\geq 0\) have range \(R\geq 1\), and suppose \(\ker H_N^{\mathrm{open}}(h)=G_N(T)\) for every sufficiently large \(N\). Let both free margins tend to infinity. Then there is one \(\gamma {\gt}0\) such that
and, for every pure \(\eta \in \mathcal F_h\) and every local observable \(X\) with \(\eta (X)=0\), the literal limit exists and satisfies
The margins may diverge along an arbitrary filter. This is the pair of conclusions in [ 225 , Theorem 1.2 ] for the stated primitive family; the original range has no injectivity lower bound. General periodic presentations remain separate [ 61 ] .
The sector gap theorem supplies one constant for both conclusions. The pure-state classification identifies each pure zero-energy state with a constructed sector, to which the same literal limit and bound apply.
The preceding finite and infinite-volume conclusions also hold when the original blocks are only assumed to have full simultaneous word span at some positive length \(S\). Their normalization and invariant density matrices are derived internally. No condition \(R\geq S+1\) is required; positivity and the eventual open-chain kernel identity remain the hypotheses on the original interaction.
Normalize the blocks, preserving every block ground space and every weighted joint ground space. The eventual kernel condition transfers to this family. Its simultaneous span excludes gauge-and-phase equivalence of different blocks. Apply the preceding theorem.
Let \(T=\bigoplus _{j=0}^{b-1}\mu _jA_j\), with nonzero weights, be a finite family of pairwise gauge-phase inequivalent trace-preserving primitive tensors of positive bond dimensions. Suppose each tensor has a faithful invariant matrix. There exist \(R{\gt}0\) and a positive interaction \(h\) on \(R\) sites such that, for every \(N\geq R\),
Neither \(R\) nor a simultaneous injectivity length is supplied. The inequality \(N\geq R\) is stronger than the inclusive-interval bound \(N\geq R+1\) in [ 225 , Theorem 1.1 ] .
Choose the positive open-kernel threshold \(R\) and let \(h\) be the orthogonal projection onto \(G_R(T)^\perp \). Its open sum is the canonical range-\(R\) parent Hamiltonian. The threshold theorem gives the stated kernel equality, including the single-window volume \(N=R\).
For the primitive family of the preceding theorem, one can choose a single \(R{\gt}0\) and \(h\geq 0\) such that the open kernels equal \(G_N(T)\) for every \(N\geq R\), and hence for all sufficiently large \(N\), and
Here \(\omega _j\) is the constructed quasi-local state of \(A_j\). The pure states in \(\mathcal F_h\) are precisely the states \(\omega _j\). Purity is extremality among all normalized positive quasi-local states; no translation invariance of a state being classified is assumed. This is the stated primitive-family case of the existence assertion in [ 225 , Theorem 1.1 and its finite-family consequence ] . More general periodic GVBS presentations remain separate [ 61 ] .
The all-volume kernel bound implies the eventual kernel hypothesis. Apply the two classification theorems to the same positive canonical interaction. They characterize the entire zero-energy face and its pure states without an invariance assumption on the state.
Let \(A\) be any MPS tensor, and let \(P_A(N)\) denote the matrix of the orthogonal projection onto \(G_N(A)\). For \(L,N\geq 0\),
If \(d{\gt}0\) and \(L{\gt}0\), then for every \(a\in \mathbb Z\) and \(N\geq 0\),
No normality or translation invariance is assumed. These are coordinate consequences of the site regrouping in [ 225 , lines 825–836 ] and the blocked local spaces of [ 17 , lines 1815–1820 and 1985–1992 ] .
Evaluate the finite projection conjugacy on a configuration basis vector. This gives the matrix reindexing identity. Apply the interval observable blocking formula to obtain the completed identity.
Let \(d,L{\gt}0\) and let \(T_L\omega =\omega \circ \overline{\mathcal B}_L^{-1}\) be functional transport from the blocked chain to the original chain. For every continuous complex-linear functional \(\omega \) on the blocked algebra,
Equivalently, for every continuous complex-linear functional \(\varphi \) on the original algebra,
Both identities include \(N=0\) and arbitrary integer \(a\).
Substitute the projection observable identity into inverse precomposition. The inverse evaluation follows by applying the same identity to \(T_L^{-1}\varphi \).
For \(d,L{\gt}0\), exact support of \(T_L^{-1}\varphi \) in every interval space of \(A^{[L]}\) is equivalent to exact support of \(\varphi \) in every aligned interval space of \(A\):
The same equivalence holds when both quantifiers over lengths are restricted to \(N{\gt}0\). These identities hold for every continuous complex-linear functional; positivity and normalization are not required.
Apply the inverse evaluation identity to each pair \((a,N)\), with the same restriction on lengths on both sides.
Let \(A\) be any MPS tensor. Let \(P_A(I)\) be the projection onto its open-boundary MPS space on a finite nonempty interval \(I\). If \(I\subseteq J\), the projection \(P_A(I)\), extended by the identity outside \(I\), fixes \(P_A(J)\). Consequently, every positive normalized quasi-local state \(\varphi \) with \(\varphi (P_A(J))=1\) satisfies \(\varphi (P_A(I))=1\). Neither injectivity nor translation invariance is required. This is the restriction direction of [ 225 , Section 3, equation (3.12) ] .
Every restriction of an open-boundary MPS vector belongs to the MPS space on the smaller interval. Thus the smaller interval’s extended ground projection absorbs the larger ground projection. Positivity and Cauchy–Schwarz imply that a state supported in a projection evaluates every observable through its compression to that projection. Compressing the extended smaller projection therefore gives expectation one.
Let \(A\) be any MPS tensor, let \(L\geq 1\), and let \(\varphi \) be a positive normalized quasi-local state. Then \(\varphi (P_A([a,a+N)))=1\) for every \(a\in \mathbb Z\) and \(N\geq 1\) if and only if \(\varphi (P_A([aL,(a+N)L)))=1\) for every \(a\in \mathbb Z\) and \(N\geq 1\). No primitivity, translation invariance, or spectral gap is assumed. This gives the support-space consequence of finite blocking in [ 225 , Section 3 ] .
One implication follows by restricting the collection of intervals. For the converse, Euclidean division writes an arbitrary left endpoint as \(a=cL+b\) with \(0\leq b{\lt}L\). An aligned interval starting at \(cL\) and containing \(b+N\) blocks contains \([a,a+N)\), even when \(a{\lt}0\). Apply the contained-interval support lemma.
Let \(C\) be an MPS tensor of bond dimension \(D\), and let \((B_j)_{j\in J}\) be a finite family of tensors with the same physical dimension and bond dimensions \(D_j\). Suppose that \(V_j:\mathbb {C}^{D_j}\to \mathbb {C}^D\) satisfy \(V_j^\dagger V_j=\mathbb {1}_{D_j}\), \(\sum _jV_jV_j^\dagger =\mathbb {1}_D\), and \(C^iV_j=V_jB_j^i\). Then \(G_N(C)=\bigvee _jG_N(B_j)\) for every \(N\ge 0\), both in computational coordinates and in the physical Hilbert space.
The letterwise intertwining implies \(C^wV_j=V_jB_j^w\) for every word, including the empty word. Trace cyclicity gives
The first identity includes each sector space in the ambient space; the second gives the reverse inclusion in their sum.
Suppose that \(A\) is periodic of period \(m\) and \(p{\gt}0\). Put \(r=\gcd (m,p)\). There are nonzero cyclic projections \(P_u\) with \(\sum _uP_u=\mathbb {1}_D\) and \(P_uA^i=A^iP_{u+1}\), positive dimensions \(D_\alpha \) with \(\sum _\alpha D_\alpha =D\), and left-canonical tensors \(B_\alpha \) indexed by \(\alpha \in \{ 0,\ldots ,r-1\} \). Their support isometries satisfy
Every \(B_\alpha \) is periodic of period \(m/r\), and for every \(N\ge 0\),
The cyclic compression is that of [ 191 , Lemma 6 ] ; the local-space equality is its supported word contraction.
The step-orbit projections commute with every blocked letter and resolve the virtual identity. The displayed compression therefore intertwines letters. Apply the isometric sector resolution to obtain \(G_N(A^{[p]})=\bigvee _\alpha G_N(B_\alpha )\), and identify the latter with the open MPS space of the direct-sum tensor.
Let \(T=\bigoplus _j\mu _j A_j\), where every \(\mu _j\ne 0\). If \(A_j\) is trace preserving and has a nonzero positive invariant matrix, its constructed state \(\omega _j\) satisfies
for every \(a\in \mathbb Z\) and \(N\geq 0\). Primitivity is unnecessary.
The sector MPS space is contained in the joint space. Thus its vectors are annihilated by \(1-P_T(N)\), which has zero expectation in the sector state. Normalization gives the assertion.
Suppose finitely many pairwise inequivalent primitive tensors \(A_j\) have faithful invariant matrices, and let \(T=\bigoplus _j\mu _j A_j\) with every \(\mu _j\ne 0\). Let \(\omega _j\) be their constructed states. A state \(\varphi \) satisfies
if and only if
Among these supported states, the pure states are precisely the \(\omega _j\). No translation invariance of \(\varphi \) is assumed. This is the primitive-family support classification used in [ 225 , Theorem 1.1 and Section 3 ] ; identification of a general periodic GVBS presentation remains separate.
Full interval support implies support on every centered interval, so the centered-interval argument supplies the convex coefficients. Conversely, every sector state has joint support, and linearity preserves this property under convex combinations. A pure state in a finite convex combination equals a constituent, and every constituent is pure among all states.
Let \(A\) be periodic of period \(m\), and let \(p{\gt}0\) satisfy \(m\mid p\). Put \(r=\gcd (m,p)=m\). There are positive bond dimensions \(D_j\) with \(\sum _{j=0}^{r-1}D_j=D\), tensors \(B_j\) on the blocked physical alphabet, and positive definite matrices \(\rho _j\), such that each pair \((B_j,\rho _j)\) has primitive fixed-point data. In particular,
and for every \(N\geq 0\),
No inequivalence assumption is imposed on the sector family. This combines the periodic blocking of [ 191 , Lemma 6 ] with the primitive normalization used in [ 225 , Section 3, equations (3.1)–(3.2b) ] . The peripheral period is the corrected \(m/\gcd (m,p)\) of [ 86 ] .
The compressed sectors have period \(m/\gcd (m,p)=1\), hence are irreducible, trace preserving, and peripheral primitive. Finite-dimensional Perron–Frobenius theory supplies their primitive fixed-point data, and irreducibility makes each invariant matrix positive definite. Retain the compressed tensors themselves, so the exact direct-sum support identity is unchanged at every length.
A periodic tensor of period \(m\) admits \(m\) positive-dimensional primitive sectors after blocking by \(m\). Their bond dimensions sum to \(D\), they have faithful invariant matrices, and
Take \(p=m\) and use \(m{\gt}0\) and \(\gcd (m,m)=m\).
Gauge-equivalent MPS tensors, including a common nonzero scalar factor on every physical matrix, have the same open-boundary MPS space at every length. More generally, normalized normal tensors whose periodic vectors differ by powers of one nonzero scalar have identical open-boundary spaces at every length, even when their bond dimensions are initially allowed to differ. The latter assertion uses the dimension equality and invertible gauge supplied by the fundamental theorem [ 13 , Proposition 2.7, Theorem 2.10 ] .
An invertible similarity is absorbed into the boundary matrix, while a nonzero length-dependent scalar is absorbed into its normalization. For normalized normal tensors, the periodic phase relation first gives equal bond dimensions and a gauge relation. Apply the first assertion.
Let \((A_j)_{j=0}^{r-1}\) be a finite family of normalized primitive tensors of positive bond dimensions, with faithful fixed-point witnesses \(\rho _j\). Let \(\mu _j\neq 0\). There is an injective selection of original indices \(s:\{ 0,\ldots ,g-1\} \to \{ 0,\ldots ,r-1\} \) such that the tensors \(B_k=A_{s(k)}\) retain the supplied primitive and faithful witnesses, and distinct \(B_k\) are not gauge-phase equivalent after any equality of their bond dimensions. Every \(A_j\) is gauge-phase equivalent to one \(B_k\), with the required bond-dimension equality supplied. For every \(N\geq 0\),
The finite family may be empty. This is the exact local-support consequence of selecting phase-class representatives in [ 13 , Proposition 2.7 ] , used for the inequivalent primitive sectors of [ 225 , Section 3 ] .
Select one original index from each periodic phase class. Distinct selections are gauge-phase inequivalent. Primitivity and faithfulness imply normalized normality, so each original tensor has exactly the same local MPS spaces as its selected representative. The local space of a weighted direct sum with nonzero weights is the sum of the block spaces. Replacing repeated summands by their representatives therefore preserves that sum at every length. The witnesses are inherited because the representatives are selected original tensors.
Let \(A\) be periodic of period \(m\), and let \(p{\gt}0\) satisfy \(m\mid p\). There is a nonempty finite family \((B_j)_{j=0}^{g-1}\), with positive bond dimensions and faithful invariant matrices \(\rho _j\), such that each \((B_j,\rho _j)\) has normalized primitive fixed-point data, distinct sectors are gauge-phase inequivalent after any equality of their bond dimensions, and
The tensors and matrices are selected from the compressed primitive family. Inequivalence is derived, rather than assumed. This is a finite local-support consequence of the periodic blocking in [ 191 , Lemma 6 ] and the representative selection in [ 13 , Proposition 2.7, Theorem 2.10 ] . The compressed peripheral period uses the correction in [ 86 ] .
Obtain primitive compressed sectors with their exact joint local space. Select one representative of each gauge-phase class. The supplied primitive and faithful witnesses are inherited, and the joint support is preserved at every length. Since the original period is positive, the compressed family is nonempty; the gauge cover therefore contains at least one selected representative.
Blocking a periodic tensor by its period gives a nonempty finite pairwise gauge-phase inequivalent primitive family with faithful invariant matrices and
Apply the theorem with \(p=m\), using positivity of the period.
Let \(A\) be any MPS tensor of positive physical dimension, and let \(\varphi \) be a normalized positive norm-one quasi-local state. If \(\varphi (j_{-n,2n+1}(p_A(2n+1)))=1\) for every sufficiently large \(n\), then \(\varphi (j_{a,N}(p_A(N)))=1\) for every \(a\in \mathbb Z\) and every \(N{\gt}0\). This is the finite-support restriction used in [ 225 , Theorem 1.1 and Section 3 ] .
Choose \(n\) sufficiently large that the symmetric interval \([-n,n]\) contains \([a,a+N-1]\) and has full ground-projection expectation. Support descends to the contained interval.
Let \(A\) be any MPS tensor of positive physical dimension, and let \(h\geq 0\) be a local interaction of range \(R{\gt}0\). Suppose that \(\ker H_N(h)=G_N(A)\) for every sufficiently large \(N\). For every continuous complex-linear functional \(\varphi \),
These are the eventual finite-volume kernel and local support conditions of [ 225 , Theorems 1.1–1.2 and Section 3 ] . No translation invariance of \(\varphi \) or injectivity bound on \(R\) is assumed.
Zero local energy first gives full support on sufficiently large symmetric intervals, and hence on every nonempty interval. Conversely, fix \(a\) and choose \(N\geq R\) with the exact kernel. Set \(p=p_A(N)\) and restrict \(\varphi \) to this interval. Since \(H_N(h)p=0\) and \(\varphi (j_{a,N}(p))=1\), support compression gives \(\varphi (j_{a,N}(H_N(h)))=0\). The finite open sum is the sum of its translated interaction observables. Their expectations are all nonnegative, so each is zero; its first summand gives \(\varphi (j_{a,R}(h))=0\).
Let \(d,L{\gt}0\), let \(A\) be a tensor, and let \(\varphi \) be a normalized positive state on the original chain. Write \(T_L\) for transport of states from the blocked chain. Then
This statement requires neither primitivity nor translation invariance. It makes the regrouping in [ 225 , lines 825–836 and Section 3 ] explicit at the level of supported states.
Support on all intervals is equivalent to support on the aligned intervals. The blocked projection identity transports the latter conditions to all intervals of the blocked chain.
Suppose \(G_{A^{[L]}}(N)=G_C(N)\) for every \(N\geq 0\). Then a state \(\varphi \) is supported on all nonempty original \(A\)-intervals if and only if \(T_L^{-1}\varphi \) is supported on all nonempty \(C\)-intervals. No primitivity assumption is made.
Equal subspaces have equal orthogonal projections. Substitute these projections in the support equivalence.
Let \(d,L{\gt}0\). Suppose \(B_1,\ldots ,B_b\) are pairwise inequivalent primitive tensors of positive bond dimensions on the blocked alphabet, with faithful invariant matrices, and \(\mu _j\ne 0\). Assume the exact identities
Let \(\omega _j\) denote the normalized sector states on the blocked chain. For any normalized positive state \(\varphi \) on the original chain, support on all nonempty \(A\)-intervals is equivalent to
The pure supported states are exactly the states \(T_L\omega _j\). Purity and the support condition are taken among all states. This is the conclusion of [ 225 , Theorem 1.1, lines 854–926 ] for the specified exact representation; its existence and identification with a general GVBS construction are separate assertions.
Apply the primitive-family classification to \(T_L^{-1}\varphi \). State transport preserves finite real convex combinations and purity, so transport the resulting identities back to the original chain.
Let \(d,L,K{\gt}0\), let \(b+K\leq N\), and let \(h\) be any operator on \(K\) blocked sites. Write \(W_{N,b}(h)\) for its nonwrapping placement starting at \(b\), and \(\mathcal R_{N,L}\) for decoding both matrix indices. Then
This is an operator identity under the site regrouping in [ 225 , lines 825–836 and Section 3 ] and [ 16 , Appendix, lines 2308 and 2313–2320 ] .
Apply the injective interval inclusion to both matrices. Their quasi-local images agree by the window inclusion and interval blocking identities: the blocked start \(b\) becomes \(bL\) and the length becomes \(KL\).
For \(d,L,K{\gt}0\) and every \(N\geq 0\), write \(H_h(N)\) for the open sum of the range-\(K\) interaction \(h\) on the blocked chain. Then
The sum is zero when \(N{\lt}K\). This is the open interaction convention of [ 225 , equation (3.12) ] transported under blocking.
Matrix reindexing is linear. Apply the window identity to each term of the finite open sum.
Let \(d\geq 0\), \(K,L{\gt}0\), and let \(Y\geq 0\) be an operator on \(KL\) original sites. For every \(N\geq 0\),
The coefficient is one. This is a derived positive-term counting consequence of [ 225 , equation (3.12) ] .
The map \(b\mapsto bL\) is injective because \(L{\gt}0\), and every aligned start satisfies \(bL+KL\leq NL\). Thus the left sum is a subsum of the full open sum. Each omitted term is positive, since window placement is a star-algebra homomorphism.
Let \(A\) be a periodic MPS tensor of period \(m\) and positive physical dimension. Let \(h\geq 0\) have interaction range \(R{\gt}0\), and suppose \(\ker H_N(h)=G_N(A)\) for every sufficiently large \(N\). There is a nonempty finite family of pairwise gauge-phase inequivalent primitive tensors \(B_j\), with positive bond dimensions and faithful invariant matrices \(\rho _j\), such that \(G_N(A^{[m]})= G_N(\bigoplus _j B_j)\) at every length \(N\geq 0\). Let \(\omega _j\) denote the primitive state of \((B_j,\rho _j)\) transported to the original lattice by grouping \(m\) consecutive sites. For every continuous complex-linear functional \(\varphi \),
The sector data and their inequivalence are derived from periodicity. The state is not assumed translation invariant, and \(R\) is not subject to an injectivity bound. This is the periodic-tensor case of the ground-state face classification in [ 225 , Theorems 1.1–1.2 and Section 3 ] ; the identification of a general periodic GVBS presentation remains separate, as described in [ 61 ] .
Blocking by the positive period gives inequivalent primitive sectors with faithful invariant matrices and exact joint support at every length. The eventual kernel identity makes membership in \(\mathcal F_h\) equivalent to full expectation on every nonempty interval support projection. Transporting this condition to the blocked lattice identifies the supported states with the convex hull of the primitive sector states. The blocking equivalence preserves purity among all states, giving (448).
Let \(A_j\) be a finite family of tensors and let \(\mu _j\neq 0\). For all integers \(L,N\geq 0\), the open-boundary space of the blocked weighted sum satisfies
No periodicity, positivity, or nonempty-family assumption is required. These are the boundary spaces of [ 225 , Section 3 ] under the blocking coordinates of [ 191 , Lemma 6 ] .
The configuration isometry identifies each blocked boundary space with the original boundary space at length \(NL\). Under this isometry, the space of the weighted sum is the sum of the original block spaces. A linear isomorphism preserves sums of subspaces, giving the asserted identity in the blocked coordinates.
Let \((B_{j,a})_a\) be finitely many finite families of primitive tensors of positive bond dimensions, with faithful invariant matrices. There is a finite pairwise gauge-phase inequivalent primitive faithful family \((C_b)_b\) such that, for every \(N\geq 0\),
Any of the families may be empty. This is representative selection in [ 13 , Proposition 2.7 ] applied to a finite union of primitive families.
Index all pairs \((j,a)\) by one finite set and apply primitive representative selection. A sum over a dependent finite index set is the iterated sum over its two indices, so the selected family retains the asserted boundary space at every length.
Let \(A_j\) be a finite family of normalized periodic tensors with periods \(m_j\), and let \(\mu _j\neq 0\). Suppose \(L{\gt}0\) and \(m_j\mid L\) for every \(j\). There are finitely many primitive tensors \(B_k\) of positive bond dimensions, with faithful invariant matrices \(\rho _k\), such that distinct \(B_k\) are not gauge-phase equivalent after any equality of their bond dimensions. For every \(N\geq 0\),
The original family may be empty, and the representative family may consequently be empty. This is the finite-family support consequence of [ 191 , Lemma 6 ] and representative selection in [ 13 , Proposition 2.7 ] .
The cyclic-sector decomposition of \(A_j^{(L)}\) has compressed period \(m_j/\gcd (m_j,L)=1\), so each compressed tensor is primitive and has a faithful invariant matrix. Flatten these finitely many sector families. The joint boundary-space identity identifies their sum with the blocked original space. Selecting one representative from each gauge-phase class preserves this sum at every length and inherits the primitive and faithful witnesses.
Every finite weighted family of normalized periodic tensors with nonzero weights admits a positive blocking length and an inequivalent primitive faithful family with identical open-boundary spaces at every blocked length. No original inequivalence or nonempty-family assumption is required. This is the common-period consequence of [ 191 , Lemma 6 ] .
Take \(L=\prod _jm_j\). Every period is positive, and every \(m_j\) divides this product. For an empty family, the product equals one. Apply the common-multiple presentation theorem.
For a range-\(R\) matrix \(h\), let \(H_h(n)\) be its finite open interaction sum. Group \(L\) sites into one site and let \(\mathcal R_{K,L}\) decode both indices of a matrix on \(K\) grouped sites. Define
Decoding \(h_B\) recovers \(H_h(KL)\). The definition allows \(d,L,K\geq 0\). This is a regrouping construction for [ 225 , Section 3, equations (3.12)–(3.14) ] .
If \(R{\gt}0\) and \(h\geq 0\), then \(h_B\geq 0\), including when \(KL{\lt}R\).
Reindexing is an equivalence. Each nonwrapping local term is positive because its inclusion is a star-algebra homomorphism; finite sums and matrix reindexing preserve positivity.
Let \(d,L,R,K{\gt}0\), \(R\leq KL\), and \(N\geq K\). If \(h_B\) is the grouped interaction, then
where \(W_{n,b}\) places an operator at the nonwrapping start \(b\). This is the aligned interaction sum of [ 225 , equation (3.12) ] with each grouped interaction expanded into its original terms.
Decoding sends each grouped window to the corresponding original window. Inclusion is linear, and nested nonwrapping inclusions add their starts, so the internal start \(j\) becomes \(iL+j\).
Let \(d,L,R,K{\gt}0\), \(h\geq 0\), \(N\geq K\), and \(R+L\leq KL+1\). Then
The decoded operators have equal kernels. Equivalently, the configuration isometry maps \(\ker H_{h_B}(N)\) onto \(\ker H_h(NL)\). These are derived positive counting consequences of [ 225 , Section 3, equations (3.12)–(3.14) ] ; no local MPS-kernel condition is required.
For an original start \(s\), choose \(i=\min (\lfloor s/L\rfloor ,N-K)\) and \(j=s-iL\). The covering condition gives \(0\leq j\leq KL-R\), and the pair \((i,j)\) determines \(s\), giving the lower bound by a positive subsum. For each fixed \(j\), the starts \(iL+j\) are distinct and nonwrapping, so their sum is bounded by \(H_h(NL)\). Sum this bound over the \(KL-R+1\) internal offsets. Two-sided positive comparison gives kernel equality, and the configuration isometry transports that kernel.
Let \(d,L,R,K{\gt}0\), \(R+L\leq KL+1\), \(h\geq 0\), and let \(A\) be any MPS tensor. Suppose that \(\ker H_h(n)=\mathcal G_n(A)\) for all sufficiently large \(n\). Then, for all sufficiently large \(N\),
After decoding, this is the equality with the isometric image of \(\mathcal G_N(A^{[L]})\). No normality or primitivity hypothesis is needed. This transports the eventual kernel hypothesis of [ 225 , Theorem 1.2 ] through its regrouping convention.
Since \(L{\gt}0\), the original volumes \(NL\) tend to infinity. Apply the original eventual kernel equality and the exact isometric transport \(\mathcal G_N(A^{[L]})\mapsto \mathcal G_{NL}(A)\). Injectivity of the configuration isometry gives the equality in grouped coordinates.
Let \(R{\gt}0\) and let \(h\) be a positive semidefinite interaction on \(R\) sites. For every \(n\geq 0\), the open sum \(\sum _{a=0}^{n-R}\iota _{a}(h)\) of its nonwrapping translates on \(n\) sites is positive semidefinite; for \(n{\lt}R\) the sum is empty and equal to zero. This is the positivity of the finite-volume Hamiltonians in [ 225 , equation (3.12) ] .
Each translate is the tensor product of \(h\) with an identity, hence positive semidefinite, and a finite sum of positive semidefinite matrices is positive semidefinite.
Let \(\mathcal A\) be a star-ordered complex algebra admitting self-adjoint decomposition, and let \(f:\mathcal A\to \mathbb C\) be positive and complex linear. If \(f(C^*C)=0\), then \(f(YC^*C)=0\) for every \(Y\in \mathcal A\). This is the Cauchy–Schwarz implication used in the zero-energy argument of [ 225 , Section 6 ] .
Apply Cauchy–Schwarz to \(u=(YC^*)^*\) and \(v=C\). Its right-hand side contains \(\sqrt{|f(C^*C)|}=0\).
Let \(f\) be a positive complex-linear functional on a finite matrix algebra, and let \(H\geq 0\) satisfy \(f(H)=0\). Every left multiple of \(H\) has zero expectation, and
If \(C\geq 0\), \(c\in \mathbb R\), and \(C\leq cH\), then
These are finite-dimensional positive-comparison consequences of the zero-energy argument in [ 225 , Section 6 ] . Normalization of \(f\) is unnecessary.
Factor \(H=S^*S\) and apply the vanishing-square implication to obtain \(f(YH)=0\). Expanding the commutator proves (456). Positivity gives \(0\leq f(C)\leq c f(H)=0\), hence the same identity for \(C\). Conjugating \(C\leq cH\) by \(X\) gives \(X^*CX\leq cX^*HX\); monotonicity of \(f\) proves (457).
Let \(e:I\to J\) be a bijection of finite configuration sets, and let \(H\) be any square complex matrix on \(I\). The associated Euclidean isometry \(U_e\) satisfies
Every estimate \(\gamma \| v\| \leq \| Hv\| \) on \((\ker H)^\perp \) holds with the same constant for \(H^e\) on \((\ker H^e)^\perp \). Positivity is unnecessary.
Change the summation index in the matrix action. The intertwining identity and invertibility give the kernel identity. Isometries preserve norms and orthogonal complements, yielding the gap estimate.
Let \(d,L,R,K{\gt}0\), \(N\geq K\), \(R+L\leq KL+1\), and \(h\geq 0\). Let \(\widehat h\) be the interaction obtained by regrouping \(H_h(KL)\). If \(H_{\widehat h}(N)\) has norm gap \(\gamma {\gt}0\) on its kernel complement, then \(H_h(NL)\) has norm gap \(\gamma /(KL+1-R)\) on its entire kernel complement. This is a finite-volume consequence of [ 225 , Section 3, equations (3.12)–(3.14) ] .
Transport the blocked norm estimate to original coordinates. The two positive order comparisons give coincident kernels and transfer the gap with the stated divisor.
Let \(d,L{\gt}0\). Suppose \(A^{[L]}\) has an exact joint-space presentation at every length by inequivalent primitive tensors of positive bond dimensions, with faithful invariant matrices and nonzero weights. Let \(h\geq 0\) have positive range \(R\) and eventual original kernel \(G_A(n)\). If \(K{\gt}0\) and \(R+L\leq KL+1\), there is \(\delta {\gt}0\) such that
Only block-divisible original volumes are asserted. This is the corresponding restriction of [ 225 , Theorem 1.2 ] .
The coarse interaction is positive and has eventual joint primitive kernel. Apply the primitive-family uniform gap theorem and transfer its estimate for \(N\geq K\). Take a positive minimum with the finitely many smaller volumes, using their actual kernels.
Let \(A\) be a periodic tensor of period \(m{\gt}0\), and let \(h\geq 0\) have positive range and eventual kernel \(G_A(n)\). There exists \(\delta {\gt}0\) such that, for every \(N\geq 0\),
The primitive family, its invariant matrices, and its inequivalence are derived from periodicity. Other residue lengths remain separate.
Obtain the exact primitive presentation after blocking by \(m\) and choose \(K=R+1\). Since \(m\geq 1\), the window coverage condition holds.
Let \(d\geq 1\), and let \((A_j)_{j=0}^{r-1}\) be a finite family of normalized periodic tensors with periods \(m_j\), and let \(T=\bigoplus _j\mu _jA_j\), where \(\mu _j\neq 0\). Let \(h\geq 0\) have interaction range \(R{\gt}0\). Suppose that its open-chain Hamiltonians satisfy \(\ker H_h(N)=\mathcal G_T(N)\) for all sufficiently large \(N\). Then there are \(L{\gt}0\) and a finite pairwise gauge-phase inequivalent primitive family \((B_k)_{k=0}^{g-1}\), of positive bond dimensions with faithful invariant matrices, such that \(\mathcal G_{T^{[L]}}(N)=\mathcal G_{\bigoplus _kB_k}(N)\) for every \(N\geq 0\). Let \(\widetilde\omega _k\) be the corresponding primitive sector states transported from the blocked lattice to the original lattice. For every continuous complex linear functional \(\varphi \),
Here \(\mathcal F_h\) is the zero-energy face of all normalized positive states, and purity is extremality among all states. The original family may be empty and no inequivalence is imposed on it. No translation invariance of \(\varphi \), supplied invariant matrices, or injectivity bound on \(R\) is assumed.
This is the explicit periodic-tensor-family case of [ 225 , Theorems 1.1–1.2, Section 3 ] , using the site grouping at lines 825–836 and [ 191 , Lemma 6 ] . Identification of an arbitrary GVBS presentation with these tensor data and finite support spaces remains separate [ 61 ] .
Blocking by a common period multiple and selecting primitive representatives gives the exact boundary-space equality at every length. The eventual kernel identity identifies zero local energy with support on every original MPS interval. Transporting support through blocking identifies these states with convex combinations of the primitive sector states. Conversely, such a combination is a normalized positive state with all interval supports, hence has zero local energy. Blocking preserves the whole state space and purity, so the pure-support classification gives the second equivalence.
Let \(L{\gt}0\), and let \(B\) be a tensor on the blocked physical alphabet of \(L\) sites with \(\sum _i(B^i)^*B^i=I\), and let \(\rho \geq 0\) be a fixed point of its transfer map with \(\operatorname {tr}\rho \neq 0\). Transport the quasi-local state of \(B\) and \(\rho \) to the original chain through the blocking isomorphism. The transported state is invariant under translations by integer multiples of \(L\). On an aligned interval of \(NL\) sites it is the insertion expectation of \(B\) after reindexing the configurations into blocks, and an observable on \(k\geq 1\) sites placed inside such an interval has the insertion expectation of its nonwrapping placement [ 225 , lines 825–836 and Section 3 ] .
Translation by \(bL\) original sites corresponds to translation by \(b\) blocked sites, under which the blocked state is invariant. The interval identities follow from the coordinate form of the blocking isomorphism and the definition of the quasi-local state on intervals.
Let \(d\geq 1\), and let \(A_j\) be finitely many periodic tensors with nonzero weights \(\mu _j\), and let \(h\geq 0\) be an interaction of positive range whose open-chain kernels eventually equal the boundary spaces of \(\bigoplus _j\mu _jA_j\). There is \(L{\gt}0\) such that every pure zero-energy state of \(h\) is invariant under translation by every integer multiple of \(L\). No invariance under one-site translation and no minimality of the period is asserted [ 225 , Theorem 1.1 ] .
Every pure zero-energy state is one of the transported primitive sector states of a common blocking by \(L\), and each such state is invariant under translations by multiples of \(L\).
Let \(d,L{\gt}0\). For every observable \(X\) on an original interval \([a,a+k)\) with \(k{\gt}0\), there exist \(b\in \mathbb Z\), \(N{\gt}0\), and an observable \(Y\) on \(N\) blocked sites such that
Negative interval starts are included. This is the local observable form of the regrouping in [ 225 , lines 825–836 and Section 3 ] .
Put \(c=a\bmod L\), \(b=\lfloor a/L\rfloor \), and \(N=c+k\). Since \(L\geq 1\), the original interval lies inside \([bL,(b+N)L)\). Place \(X\) at offset \(c\) in this enlarged interval, adjoin identities, and invert the configuration reindexing. The interval placement and blocking identities give the asserted equality.
Let \((A_j)\) be a finite family of pairwise gauge-phase inequivalent tensors, of bond dimensions \(D_j\). If \(i\neq j\) and \(D_j=D_i\), then the tensor \(A_j\), transported by this bond-space identification, is not gauge-phase equivalent to \(A_i\). No normalization, positive dimension, or nonempty-family assumption is required. This is the separation condition of [ 13 , Proposition 2.7 ] written in the bond-space orientation used by the sector support and gap arguments.
Apply the separation condition to the ordered pair \((j,i)\) and the supplied dimension equality. The two descriptions of the bond-space identification are identical.
Let \(A\) have physical dimension \(d\) and bond dimension \(D\), let \(B\) have blocked physical dimension \(d^L\) and bond dimension \(E\), and let \(V\colon \mathbb {C}^E\to \mathbb {C}^D\). For \(N,r\geq 0\), define a linear boundary map on \(M_{D\times E}(\mathbb {C})\) by
where \(u\) is the \(N\)-letter blocked prefix and \(\tau \) is the \(r\)-letter original tail. Its physical realization lies on \(NL+r\) original sites. The Euclidean realization uses Hilbert–Schmidt virtual coordinates, with the column index preceding the row index. All dimensions and lengths may be zero. These are the residual word contractions under the compression in [ 191 , Lemma blocking-arbitrary ] and the local supports of [ 225 , Section 3, lines 1504–1538 ] .
Linearity follows from matrix multiplication and trace. Frobenius vectorization and the physical \(\ell ^2\) identification preserve the resulting vector.
If \(V^\dagger (A^{[L]})^i=B^iV^\dagger \) for every blocked letter \(i\), then every \(N,r\geq 0\) and \(Y\in M_{D\times E}(\mathbb {C})\) satisfy
This is the literal boundary-vector identity from compressed words, including empty prefixes and tails.
The letter intertwiner extends to every blocked word, so \(B^uV^\dagger =V^\dagger A^{\widetilde u}\). Word concatenation and cyclicity of the rectangular trace give \(\operatorname{tr}(B^uV^\dagger A^\tau Y) =\operatorname{tr}(A^{\widetilde u}A^\tau YV^\dagger )\).
Let \(B_a,V_a\) be a finite family satisfying \(\sum _aV_aV_a^\dagger =I\) and \(V_a^\dagger (A^{[L]})^i=B_a^iV_a^\dagger \) for every \(a,i\). Then, for every \(N,r\geq 0\),
The same equality holds in the physical Euclidean space. No orthogonality or injectivity of the residual ranges is assumed.
Resolving the right virtual boundary gives \(\Gamma ^A_{NL+r}(X)=\sum _a\mathcal F_{A,B_a,V_a}^{N,r}(XV_a)\). Conversely, each residual vector is the original boundary vector \(\Gamma ^A_{NL+r}(YV_a^\dagger )\). The Euclidean identifications transport ranges and their finite sum.
Let \(A\) be a normalized irreducible tensor of period \(m\) and let \(L{\gt}0\). There exist sector dimensions \(D_a{\gt}0\), tensors \(B_a\) on \(d^L\) spins, and isometries \(V_a\colon \mathbb {C}^{D_a}\to \mathbb {C}^D\), indexed by \(a\in \{ 0,\ldots ,\gcd (m,L)-1\} \), such that \(\sum _aD_a=D\), \(\sum _aV_aV_a^\dagger =I\), and each \(B_a\) is trace-preserving with period \(m/\gcd (m,L)\). Both letter intertwinings hold:
For every \(N,r\geq 0\), including zero, \(\mathcal G_{NL+r}(A)=\sum _a\operatorname {ran} \mathcal F_{A,B_a,V_a}^{N,r}\). This extends the finite support decomposition in [ 191 , Lemma blocking-arbitrary ] to an ungrouped terminal interval. The period formula includes the peripheral-root correction in [ 86 ] .
Step-orbit compression gives the normalized tensors, support isometries, resolution, and right intertwining. The orbit projections commute with every blocked letter. Using \(V_a^\dagger V_a=I\) gives the left intertwining. Apply the residual range resolution to this derived data.
Let \(d\geq 1\), and let \((A_j)\) be a finite family of normalized periodic tensors with periods \(m_j\), and let \(T=\bigoplus _j\mu _jA_j\), with \(\mu _j\neq 0\). Let \(L{\gt}0\) satisfy \(m_j\mid L\) for every \(j\). Suppose \(h\geq 0\) has range \(R{\gt}0\) and \(\ker H_h(n)=\mathcal G_T(n)\) for all sufficiently large \(n\). There is \(\gamma {\gt}0\) such that, for every \(N\geq 0\) and every \(v\perp \ker H_h(NL)\),
This includes short volumes with their actual kernels. The original family may be empty; its sectors need not be inequivalent and no faithful invariant matrices are supplied. This is the explicit periodic-family, block-divisible finite-chain consequence of [ 225 , Theorem 1.2, Section 3, equations (3.12)–(3.14) ] and [ 191 , Lemma 6 ] . The other original residue lengths and general GVBS presentations remain separate [ 61 ] .
Derive an exact primitive faithful inequivalent presentation after blocking by \(L\). Take the coarse interaction range \(K=R+1\); positivity of \(L\) gives \(R+L\leq KL+1\). The positive window comparison transfers the primitive coarse gap to every sufficiently long original volume of length \(NL\). Taking a finite minimum with the positive gaps on the finitely many remaining kernel complements gives one constant for all \(N\).
Assume \(d\geq 1\). Under the periodic-family, nonzero-weight, positivity, positive-range, and eventual original kernel hypotheses of Theorem 12.9.166, there exist \(L{\gt}0\) and \(\gamma {\gt}0\) such that the same norm gap holds at every original length \(NL\), on the orthogonal complement of its actual kernel. No common blocking length is supplied, and the original family may be empty. These are the aligned finite-chain lengths associated with the regrouping in [ 225 , lines 825–836, Theorem 1.2 ] .
The product \(L=\prod _jm_j\) is positive and divisible by every period. For an empty family it equals one. Apply the common-multiple theorem.
Let \(h\) have positive range \(R\), and let \(X,Y\) be observables on positive-length intervals with the same image in the quasi-local algebra. The images of their stabilized local commutator energies \(X^*[H,X]\) and \(Y^*[H,Y]\) are equal.
Place both observables in a common interval containing the interaction margins on either side. The finite-volume commutator energy is the same for the two representations. Locality identifies it with each stabilized representative.
Let \(\varphi \) be a continuous complex-linear functional that is nonnegative on squares, and let \(h\geq 0\) have range \(R{\gt}0\). If \(\varphi \) vanishes on every translate of \(h\), then every stabilized local commutator energy has real, nonnegative expectation. Normalization and translation invariance are not required.
On the enlarged finite interval, write the interaction sum as \(H\geq 0\) and restrict the functional to its matrix algebra. Its value on \(H\) is zero, so \(\varphi (X^*[H,X])=\varphi (X^*HX)\geq 0\).
Let \(d,L,R,K{\gt}0\), let \(h\geq 0\) have range \(R\), and assume \(R+L\leq KL+1\). On the grouped chain, let \(h_B\) be the original open interaction sum on \(KL\) sites expressed in grouped coordinates, and put \(M=KL+1-R\). Let \(\varphi \) be positive on squares and vanish on every original translate of \(h\), and let \(\psi =\varphi \circ \mathcal B_L\) be its functional on the grouped quasi-local algebra. For every grouped interval observable \(Y\), with original image \(X=\mathcal B_L(Y)\),
Both commutators denote their stabilized finite-range representatives.
In a sufficiently large grouped interval, the decoded Hamiltonian \(C\) satisfies \(0\leq C\leq MH\). The original functional has zero expectation on \(H\). Positive comparison therefore bounds the two finite commutator energies. Locality identifies their stabilized representatives.
Under the preceding hypotheses, suppose that a real constant \(\gamma \) satisfies \(\gamma \psi (Y^*Y)\leq \psi \bigl(Y^*[H_B,Y]\bigr)\) for every positive-length grouped interval observable \(Y\) with \(\psi (Y)=0\). Then every original local observable \(X\) with \(\varphi (X)=0\) satisfies
The original commutator energy is also the limit of the literal finite-volume commutator energy as both boundary distances tend to infinity, along any filter. In particular, a positive grouped gap yields a positive original gap. This is a conditional transport statement; it does not assert the existence of a gap for an arbitrary interaction. The argument is the site-grouping reduction used in [ 225 , Section 6 ] .
Every original interval observable has a grouped interval representation. Apply the grouped gap and the energy comparison to this representation. The variance is preserved by the star-algebra isomorphism, and the local energy is independent of its interval representation. Divide by \(M{\gt}0\). Locality makes the literal finite-volume energy eventually equal to the fixed representative; positivity makes both sides of the inequality real.
Let the physical dimension be positive and let \(A\) be a periodic MPS tensor. Let \(h\geq 0\) have arbitrary positive range \(R\), and suppose that for all sufficiently large \(N\) the open Hamiltonian has kernel \(G_N(A)\). There is a constant \(\gamma {\gt}0\) common to every pure state \(\varphi \) in the zero-energy face of \(h\) such that, for every positive-length local observable \(X\) with \(\varphi (X)=0\),
Here \(H_{\ell ,r}\) is the original open interaction sum on the interval extending \(\ell \) sites to the left and \(r\) sites to the right of the support of \(X\). The limit exists as a complex number along every filter for which both margins tend to infinity. The same \(\gamma \) applies to every such filter and pair of margins. Both sides are real, and the inequality uses their usual real order.
The primitive sectors and their faithful invariant matrices are derived from periodicity. No translation-invariance assumption is imposed on \(\varphi \). This proves the infinite-volume conclusion for an explicit periodic tensor presentation. The general GVBS presentation of [ 225 , Theorem 1.2 ] requires a separate identification with these finite support spaces; see [ 61 ] .
Group sites by the period. The periodic decomposition, followed by removal of repeated sectors, gives finitely many inequivalent primitive tensors with faithful invariant matrices and the same joint support at every length. Choose a sufficiently long coarse interaction, equal in grouped coordinates to the original interaction sum on its expanded interval. Positive comparison preserves its eventual kernel identity.
The primitive-sector theorem gives one positive coarse gap. The classification of pure zero-energy states identifies every original pure state with a transported primitive sector state. Gap transport gives the original constant \(\gamma _B/(KL+1-R){\gt}0\), uniformly over these sectors and all original local observables. Locality identifies the literal finite-volume energy limit.
Fixing a tail configuration gives a prefix ground vector with boundary \(V^*A^\tau Y\). Sum the prefix inner products over the tail configurations and apply Cauchy–Schwarz. The sums of the squared fiber norms are exactly the squared norms of the full vectors.
For two inequivalent normalized primitive blocked-prefix tensors of positive bond dimensions with faithful invariant matrices, their correlated residual ranges become uniformly orthogonal as \(N\to \infty \). The estimate is uniform in the original tail length \(r\); the original tensors supplying the tails need no normalization assumption.
Apply primitive prefix-space overlap decay and the preceding fiberwise estimate, which preserves the overlap coefficient.
Let \(B_j\) be a finite family of pairwise gauge-phase inequivalent normalized primitive blocked-prefix tensors of positive bond dimensions with faithful invariant matrices. Let \(F_j(N,r)\) be their correlated residual ranges. Along any filter for which \(N\to \infty \), and with arbitrary varying \(r\), one has
The finite family of residual ranges is asymptotically orthogonal by the uniform overlap estimate. Apply the finite-family projection comparison.
Let \(d,L{\gt}0\) and let \(A\) be an MPS tensor. Suppose that a finite, possibly empty, family of pairwise gauge-phase inequivalent normalized primitive tensors \(B_j\) of positive bond dimensions has faithful invariant matrices, nonzero weights \(\mu _j\), and the exact identities
Let \(h\geq 0\) have range \(R{\gt}0\), and suppose \(\ker H_h(n)=\mathcal G_n(A)\) for all sufficiently large \(n\). There is one \(\gamma {\gt}0\) with the following property. For every pure zero-energy state \(\varphi \), every \(a\in \mathbb Z\), and every matrix \(X\) on \(k{\gt}0\) consecutive original sites, put \(X_a=j_{a,k}(X)\) and assume \(\varphi (X_a)=0\). Along every filter for which the two margins \(\ell ,r\) tend to infinity, there is \(e\in \mathbb C\) such that
The energy and variance are real and nonnegative. The same \(\gamma \) works for all states, observables, filters, and diverging margins. No translation invariance of \(\varphi \) or short-range injectivity bound is assumed. This is the exact-presentation version of the blocking reduction in [ 225 , Theorem 1.2 and Section 6 ] ; identification with a general GVBS presentation remains separate, as recorded in [ 61 ] .
Take \(K=R+L\) and group the original Hamiltonian on \(KL\) sites. Coverage gives \(R+L\leq KL+1\), so its eventual kernel is the supplied joint primitive space. Obtain one positive coarse gap for all sectors. The eventual original kernel identifies every pure zero-energy state with a transported sector. Positive comparison transfers the coarse inequality with divisor \(KL+1-R{\gt}0\). The coarse constant is chosen before any state, observable, or exhaustion. Finite-range locality gives the literal limit and positivity gives its reality.
Let \(d{\gt}0\), let \((A_j)\) be a finite, possibly empty, family of normalized periodic tensors of periods \(m_j\), and let \(\mu _j\neq 0\). Put \(T=\bigoplus _j\mu _jA_j\). Let \(h\geq 0\) have arbitrary range \(R{\gt}0\), and suppose that \(\ker H_h(n)=\mathcal G_n(T)\) for every sufficiently large \(n\). There is one \(\gamma {\gt}0\) such that every pure zero-energy state and every centered original local observable satisfy (470), for every filter and every pair of diverging boundary margins. The common blocking length, primitive tensors, faithful invariant matrices, and sector inequivalence are derived from periodicity. Competing states are not assumed translation invariant. For an empty tensor family the zero-energy state face is empty. This concerns explicit periodic tensor families; the general GVBS identification in [ 225 , Theorem 1.2 ] remains separate; see [ 61 ] .
Block by the product of the positive periods, which equals one for an empty family. Decompose the blocked sectors, remove gauge-phase repeats, and obtain an exact joint primitive presentation at every length. Apply the preceding theorem to the original tensor \(T\) and interaction \(h\).
Let \(A\) be periodic with period \(m\). There are \(m\) primitive normalized compressed tensors \(B_j\), positive virtual dimensions \(D_j\), support isometries \(V_j\), and faithful invariant matrices \(\rho _j\). The tensors \(B_j\) are pairwise gauge-phase inequivalent and satisfy
Let \(F_j(N,r)\) be the correlated rectangular boundary range with coefficient \(\operatorname {tr}(B_j^uV_j^*A^\tau Y)\). Then
for every \(N,r\geq 0\). For every function \(r:\mathbb N\to \mathbb N\),
All primitive normalization and separation data are derived from periodicity; none are assumed separately. This comparison does not assert an overlapping-window defect estimate or a gap at all residue lengths.
Choose the primitive cyclic sector resolution, including its faithful invariant matrices, pairwise separation, and both intertwiners. The support projections resolve the virtual identity, which gives the exact residual support decomposition. Apply the residual-sector projection-sum limit and substitute this decomposition.
For residual boundary vectors \(\Gamma _{N,r}^{A,B,V}(Y)\) and \(\Gamma _{N,r}^{A,B,V}(Z)\), one has
This includes zero prefix and tail lengths and requires no normalization or letter intertwining.
Decode the blocked prefix and split the physical configuration into prefix and tail. Fixing the tail gives exactly the displayed prefix contraction; sum the resulting scalar inner products over both configurations.
Let \(T:E\to \mathcal H\) be a continuous linear map from a finite-dimensional Hilbert space to a complete Hilbert space. Suppose \(Q=Q^*\), \(TQ=T\), and \((T^*T)H=Q\). Then \(P_{\operatorname {ran}T}=THT^*\).
More generally, let \(T:E\to \mathcal H\), \(L:F\to \mathcal H\), and \(C:G\to \mathcal H\) have finite-dimensional domains and a common factorization \(C=TJ_1=LJ_2\). For \(S\in \{ T,L,C\} \), suppose \(Q_S=Q_S^*\), \(SQ_S=S\), and \((S^*S)H_S=Q_S\). If also \(H_T=H_T^*\), then
This is the supported-boundary algebra for the interval projector comparison in [ 225 , Lemma 6.1(ii) ] .
The vector \(THT^*x\) lies in \(\operatorname {ran}T\). Taking adjoints of \(TQ=T\) gives \(QT^*=T^*\), so \(T^*(x-THT^*x)=T^*x-QT^*x=0\). Thus \(THT^*\) is the orthogonal range projection. For the three-map identity, self-adjointness of \(H_T\) gives \(H_T(T^*T)=Q_T\). Consequently \(TH_T(T^*T)=T\) and \((L^*L)H_LL^*=L^*\). Expanding the right-hand side of (471) and using \(TJ_1=C\) and \(J_2^*L^*=C^*\) gives \(P_{\operatorname {ran}T}P_{\operatorname {ran}L}-CH_CC^*\).
For matrices \(P_u\) indexed by \(\mathbb Z/m\mathbb Z\), define
If \(P_uA_i=A_iP_{u+1}\), both \(A_iX\) and \(XA_i\) belong to \(V_n\) whenever \(X\in V_{n+1}\). When the \(P_u\) are Hermitian, \(X\in V_n\) implies that \(X^*\) has the word degree \(n\), namely \(P_uX^*=X^*P_{u+n}\). These are the cyclic coordinates of [ 191 , Lemma 5 ] .
Substitute the cyclic letter identity in each product. For the right product, use the boundary relation at \(u-1\). Taking adjoints exchanges the two degree conventions.
Let \(m{\gt}0\), let \(P_uP_v=\delta _{uv}P_u\), and assume \(\sum _uP_u=I\). If every length-\(n\) word satisfies \(P_uA^\sigma =A^\sigma P_{u+n}\), then
In particular every vector in \(G_n(A)\) has a representative in \(V_n\). This is the cyclic-boundary preparation for the intersection property in [ 225 , equations (3.10)–(3.11) ] .
Orthogonality gives both sides of the boundary relation for \(C_n(X)\). Cycle \(P_u\) through the trace, use the word relation and idempotence, and sum the resulting terms with \(\sum _uP_u=I\).
Let \(S\subseteq \mathcal M_D\) be a complex linear subspace. If \(X^*\) belongs to the span of the length-\(n\) words for every \(X\in S\), then \(\Gamma _n\) is injective on \(S\).
For \(X,Y\in S\) with \(\Gamma _n(X)=\Gamma _n(Y)\), the trace pairing of \(X-Y\) with every length-\(n\) word vanishes. Test against \((X-Y)^*\); then \(\operatorname {tr}((X-Y)^*(X-Y))=0\), so \(X=Y\).
Suppose \(P_uA_i=A_iP_{u+1}\) and \(\sum _iA_i^*A_i=I\). If every vector of \(G_{n+1}(A)\) has a boundary representative in \(V_{n+1}\), and \(\Gamma _n\) is injective on \(V_n\), then
In particular these hypotheses follow when \(m{\gt}0\), the \(P_u\) are Hermitian orthogonal projections summing to \(I\), and the length-\(n\) word span equals \(\{ Z:\ P_uZ=ZP_{u+n}\text{ for every }u\} \). This is an algebraic criterion for [ 225 , equations (3.10)–(3.11) and Lemma existenceinteraction ] ; eventual spanning for a periodic tensor is a separate step.
Write the two sets of one-site restrictions with boundaries \(Y_i,Z_j\) in \(V_{n+1}\). Their overlap gives \(\Gamma _n(A_jY_i)=\Gamma _n(Z_jA_i)\). Both boundaries belong to \(V_n\), hence \(A_jY_i=Z_jA_i\). The left-canonical identity reconstructs the enlarged boundary as \(\sum _jA_j^*Z_j\). Under the stated spanning condition, cyclic compression supplies the representatives, and the adjoint-span criterion supplies injectivity.
For a family \((P_u)_{u\in \mathbb Z/m\mathbb Z}\), set
If \(P_uA_i=A_iP_{u+1}\), every word of length \(n\) belongs to \(W_n\). The family need not consist of projections for this conclusion. These degrees describe the cyclic sectors of [ 191 , Lemma 5 ] .
Induct on the word length and move each \(P_u\) through the first letter.
Suppose every \(P_u\) is Hermitian. If \(X\in W_{q+r}\) and \(Y\in W_r\), then \(XY^*\in W_q\).
Take the adjoint of the relation for \(Y\) at \(u+q\) and multiply it by \(X\), using the relation for \(X\) at \(u\).
Suppose that every \(P_u\) is Hermitian, \(P_uA_i=A_iP_{u+1}\), and \(\sum _iA_i^*A_i=I\). If the words of length \(q\) span \(W_q\), then the words of length \(N\) span \(W_N\) for every \(N\ge q\). This algebraic implication supplies the later lengths in the intersection argument of [ 225 , equations (3.10)–(3.11) ] ; the initial spanning equality is assumed here.
Write \(N=q+r\). The left-canonical identity implies \(\sum _{|w|=r}A_w^*A_w=I\). For \(X\in W_{q+r}\), reconstruct
Every coefficient \(XA_w^*\) belongs to \(W_q\), hence is a combination of words of length \(q\). The displayed sum is therefore a combination of words of length \(q+r\). The reverse inclusion follows from the cyclic degree of each word.
For rectangular matrices \(V_j\) with a common row space, the sector embedding sends a family \(M=(M_j)_j\) of square matrices to \(\mathcal E(M)=\sum _jV_jM_jV_j^*\).
Let \(C,B_j\) be tensors on the same physical alphabet and let \(V_j\) be rectangular matrices satisfying \(\sum _jV_jV_j^*=I\) and \(C_iV_j=V_jB_{j,i}\). Let \(\mathcal E\) be the sector embedding of Definition 12.9.188. Then every simultaneous word tuple maps to the corresponding word of \(C\). If the length-\(q\) simultaneous word span is the full product matrix algebra, \(\operatorname {ran}\mathcal E\) is contained in the length-\(q\) word span of \(C\). If also \(V_j^*V_j=I\), every matrix \(X\) commuting with all \(V_jV_j^*\) satisfies
and hence belongs to that word span. No positivity of the sector dimensions or tensor normalization is assumed. These are the embedded cyclic-sector identities of [ 191 , Lemma 5 ] .
Intertwining propagates along words. Resolve the virtual identity to identify the image of every word tuple. Taking linear spans gives the range inclusion. For a commuting matrix, each compressed term equals \(XV_jV_j^*\); summing reconstructs \(X\).
Let \(m{\gt}0\), let \(P_uA_i=A_iP_{u+1}\), and suppose that the tensor \(A^{[m]}\) has an isometric sector resolution \(B_j,V_j\), with \(j\in \{ 0,\ldots ,m-1\} \), satisfying
Suppose each support \(V_jV_j^*\) is one of the matrices \(P_u\); their cyclic labels need not be prescribed. If the length-\(q\) simultaneous word span of \(B_j\) is full, then
This supplies the initial graded spanning step for the intersection construction in [ 225 , equations (3.10)–(3.11) and Section 6 ] .
Every original word has its length as cyclic degree. Conversely \(qm\) is zero modulo \(m\), so a matrix of this degree commutes with every sector support. Apply the embedded-sector lemma, then identify the length-\(q\) blocked word span with the original length-\(qm\) span.
Let \(A\) be a normalized irreducible tensor of period one with positive bond dimension. There is a positive-definite matrix \(\rho \) such that \(A\) and \(\rho \) form a primitive normalized MPS pair. This is the Perron–Frobenius normalization used in [ 225 , Equations (3.1)–(3.2b) ] .
Period one gives a primitive trace-preserving transfer map and irreducibility of its Kraus family. The primitive Perron–Frobenius theorem supplies an invariant matrix \(\rho \); irreducibility makes it positive definite.
Let \(T:E\to F\), \(L:G\to F\), and \(C:W\to F\) be bounded maps, with finite-dimensional virtual spaces and a complete physical Hilbert space. Suppose
Then
For a family along any filter, assume this factorization eventually holds, the product \(\| T\| \| H_T\| \| H_L\| \| L\| \) is eventually bounded, and \(\| R\| \to 0\). Then \(\| P_TP_L-P_C\| \to 0\). The spaces may vary with the index. This implication supplies the norm comparison in [ 225 , Lemma commutation (ii) ] ; the factorization, inverse bounds, and residual decay are explicit hypotheses here.
Apply submultiplicativity of the operator norm and \(\| L^*\| =\| L\| \). The eventual bound then gives a constant multiple of \(\| R\| \); squeeze the nonnegative projection norm between zero and this quantity.
Expand the Hilbert–Schmidt inner products and sum the trace pairings. The residual physical inner product splits into the same prefix fibers. The primitive limiting metric acts by right multiplication, giving the stated pullback formula.
Transport \(P_a\) through each word, then use \(\sum _\tau A^{\tau \dagger }A^\tau =I\). The complement \(I-Q_r\) is positive, so \(\operatorname{tr}(Y^\dagger Q_rY)\leq \| Y\| _{\rm HS}^2\). The limiting action follows from the exact pullback metric.
Subtract the two correlated pullback forms. The fiberwise difference has norm at most \(\| K_N^B-K_\rho \| \), and the two outer correlated maps have norm at most one. Apply the primitive prefix Gram bound.
Compress the step-orbit sectors after blocking by \(m\). Their period is one, so irreducibility and trace-preserving normalization give faithful primitive fixed points. Each step orbit consists of its individual cyclic projection. Apply the cyclic limiting-action identity and the uniform primitive residual Gram estimate.
Let \(A\) have physical dimension \(d\) and bond dimension \(D\), let \(B\) have physical dimension \(d^L\) and bond dimension \(E\), and let \(V\in \mathbb {C}^{D\times E}\). For nonnegative integers \(K,M,r\), use physical configurations \((u,v,\tau )\) of respective lengths \(K,M,r\), with the first two in the blocked alphabet. Write \(\mathbf x=(X_u)_u\) for a family in \(\mathbb {C}^{D\times E}\) and \(\mathbf y=(Y_\tau )_\tau \) for a family in \(M_{E}(\mathbb {C})\). Define
The family coordinates are ordered as \((u,\text{column},\text{row})\). Decoding the blocked letters and concatenating \(u,v,\tau \) gives a bijection with the original configurations of length \((K+M)L+r\), including zero lengths. These are the residual coordinates for the overlap argument in [ 225 , Lemma 6.1(ii) ] .
With the coordinates of Definition 12.9.197, put \((J_2Y)_\tau =V^*A^\tau Y\). Then \(TJ_1=C=L_1J_2\). \(C\) is the residual boundary map at prefix length \(K+M\) and tail length \(r\), transported by the physical configuration bijection. No injectivity, normalization, or intertwining hypothesis is required.
Trace cyclicity gives \(\operatorname{tr}(B^vV^*A^\tau YB^u)=\operatorname{tr}(B^uB^vV^*A^\tau Y)\); associativity gives the second factorization. The physical-coordinate identity follows from the word product under concatenation.
In the same coordinates, let \(G_B(M)=\Gamma _B(M)^*\Gamma _B(M)\) be the ordinary prefix Gram. With the Frobenius inner product on virtual matrices,
This identity is exact at every nonnegative length; it asserts no limiting metric or quantitative decay.
Expand the physical inner product over \((u,v,\tau )\) and interchange the \(v\) and \(\tau \) sums. Trace cyclicity rewrites the left-window coefficient as \(\operatorname{tr}(B^vY_\tau B^u)\). The \(v\) sum is then the Gram pairing of \(\Gamma _B(M)\).
Let \(A\) be a normalized irreducible periodic tensor of period \(m\). There are positive sector dimensions, primitive normalized tensors \(B_j\) on blocks of \(m\) sites, faithful invariant matrices \(\rho _j\), and isometries \(V_j\) whose mutually orthogonal support projections sum to the identity. Distinct \(B_j\) are inequivalent up to similarity and phase. In the inverse cyclic labelling, the projections satisfy \(P_{j+1}A_i=A_iP_j\). Both rectangular intertwiners hold: \(A^{[m]}_iV_j=V_jB_{j,i}\) and \(V_j^*A^{[m]}_i=B_{j,i}V_j^*\). This retains the cyclic coordinates of [ 191 , Lemma 5 ] used in the grouping argument of [ 225 , Section 6 ] .
Take the cyclic isometric resolution of the periodic tensor. Each compressed blocked sector has period one, hence admits a faithful primitive invariant matrix. The cyclic sector separation theorem gives pairwise inequivalence. Multiplying the corner identity by the isometry and its adjoint gives the two intertwiners.
Let \(A\) be a normalized irreducible periodic tensor of period \(m\). There is an orthogonal resolution \(\{ P_u\} _{u\in \mathbb {Z}/m\mathbb {Z}}\) satisfying \(P_uA_i=A_iP_{u+1}\), and a threshold \(n_0\), such that for every \(n\ge n_0\) the length-\(n\) words span precisely \(\{ X:P_uX=XP_{u+n}\text{ for all }u\} \). Consequently,
The lengths here count original sites, with no divisibility requirement. This is the cyclic-sector form of the intersection argument in [ 225 , equations (3.10)–(3.11) ] .
Simultaneous spanning of the primitive blocked sectors gives spanning of one degree-zero cyclic corner algebra. Negating the cyclic labels puts the projections in the word-degree convention. The adjoint-word resolution propagates this span to all later lengths. Restricted boundary injectivity then gives the intersection identity.
For the same tensor \(A\), there is \(R_0{\gt}0\) such that, for every \(R\ge R_0\) and every \(N\ge R\), the canonical range-\(R\) open parent Hamiltonian has kernel exactly \(G_N(A)\). In particular, there exists a positive interaction \(h\) of some range \(R{\gt}0\) with this kernel identity at every \(N\ge R\). No faithful invariant matrix, cyclic resolution, or short-range intersection is supplied as a hypothesis. This concerns the explicit normalized periodic tensor presentation; it does not assert a presentation theorem for arbitrary GVBS states. It implements the finite-support interaction construction of [ 225 , Lemma existenceinteraction and Theorem 1.1 ] in this tensor setting.
Choose the interaction range beyond the intersection threshold. Iteration over contiguous intervals identifies the simultaneous local constraints with the full boundary-condition space. Positivity of the local complementary projection identifies these constraints with the open Hamiltonian kernel, including the single-window volume \(N=R\). Writing that projection in physical matrix coordinates gives \(h\).
Let \(h_R\) be the configuration-basis matrix of the orthogonal projection onto \(G_R(A)^\perp \). This matrix is positive semidefinite and represents the same projection on the Euclidean realization of the physical configuration space. It is the canonical choice in [ 225 , Lemma existenceinteraction ] .
Identify matrices with bounded operators in the configuration basis. Under this identification, matrix positive semidefiniteness is equivalent to positivity of the represented operator. Orthogonal projections are positive.
Let \(A\) be a normalized periodic tensor. There exist \(R{\gt}0\), a positive semidefinite interaction \(h\) on \(R\) sites, and \(\gamma {\gt}0\) such that \(\ker H_N(h)=G_N(A)\) for every \(N\ge R\). If \(\varphi \) is any pure zero-energy state for this interaction, and \(X\) is an observable on an interval of positive length with \(\varphi (X)=0\), then
along every filter for which both interval margins \(\ell ,r\) tend to infinity. The same constant works for all such states, observables, and exhaustions. The physical dimension, interaction, eventual kernel identity, and primitive invariant matrices are derived. This proves the existence and infinite-volume gap conclusions of [ 225 , Theorems 1.1–1.2 ] for an explicit normalized periodic tensor presentation. Identification with a general GVBS presentation, and the all-residue finite-volume gap, remain separate.
Choose the positive canonical interaction from the original-chain intersection. Its kernel identity for every \(N\ge R\) gives the eventual kernel hypothesis of the periodic infinite-volume gap theorem. Apply that theorem to obtain one positive constant before all states and interval exhaustions are chosen.
Let \(S,I,J\) be finite sets and let the scalar field be \(\mathbb {R}\) or \(\mathbb {C}\). For a vector \(x\in \ell ^2(S\times I)\), its \(s\)th fiber is \(x_s=(x_{s,i})_{i\in I}\). For a linear operator \(G:\ell ^2(I)\to \ell ^2(J)\), define \(G^{\oplus S}\) by \((G^{\oplus S}x)_s=Gx_s\). The fiber projection itself is defined without a finiteness assumption.
Finite direct sums preserve adjoints, composition, identity operators, and operator differences. They satisfy \(\lVert G^{\oplus S}\rVert \leq \lVert G\rVert \) and \(\langle y,G^{\oplus S}x\rangle =\sum _s\langle y_s,Gx_s\rangle \). These statements include empty coordinate sets.
The squared norm is the sum of the squared fiber norms. Applying \(\lVert Gx_s\rVert \leq \lVert G\rVert \lVert x_s\rVert \) to every fiber gives the norm bound. The inner-product identity follows by splitting the sum over \(S\times I\). Apply the adjoint identity in each fiber to obtain the adjoint formula. Composition, identity, and subtraction hold coordinatewise.
Use the residual windows \(T,L_1\) at full length \((K+M)L+r\) from Definition 12.9.197. Write \(\Gamma _{M,r}\) for the correlated residual boundary map and \(G_B(n)\) for the ordinary blocked-prefix Gram. Then
Consequently \(\lVert T\rVert \leq \lVert \Gamma _{M,r}\rVert \) and \(\lVert L_1\rVert \leq \lVert \Gamma _B(K+M)\rVert \). These identities require no normalization or injectivity and include zero lengths and empty configuration sets.
Expand the physical inner products over \((u,v,\tau )\). For the right Gram, first sum over the residual physical variables \((v,\tau )\), retaining \(u\) as a spectator. For the left Gram, concatenate \(u,v\) and retain \(\tau \) as a spectator. Apply the finite direct-sum norm bound to each Gram and use \(\lVert F^*F\rVert =\lVert F\rVert ^2\).
Identify each correlated tail fiber and apply the left window factorization.
Let \(\rho \in M_{E}(\mathbb {C})\) have nonzero trace, put \(G_\infty =\operatorname {gramReshuffle}(\mathcal P_\rho )\), and let \(K_r=C_r^*G_\infty ^{\oplus \tau }C_r\). If \(\lVert C_r\rVert \leq 1\), then
The second bound does not require the contractivity assumption. Neither bound requires primitivity or a fixed-point equation. Both coefficients are one and are independent of the spectator lengths.
Finite direct sums preserve differences and do not increase operator norm. For the right window also apply the coefficient-one correlated Gram bound.
On \(M_{D\times E}(\mathbb {C})\) with its Hilbert–Schmidt inner product, define \(R_\rho Y=Y\rho /\operatorname{tr}\rho \) and \(\mathcal QY=QY\). If \(E{\gt}0\) and \(\rho {\gt}0\), then \(R_\rho \) is self-adjoint and invertible, with \(R_\rho ^{-1}Y=(\operatorname{tr}\rho )Y\rho ^{-1}\). If \(Q\) is an orthogonal projection, then so is \(\mathcal Q\). Right multiplication commutes with every row operator \(\mathcal Q\).
In column-first rectangular coordinates the operators are \((\operatorname{tr}\rho )^{-1}(\rho ^{\mathsf T}\otimes I_D)\) and \(I_E\otimes Q\). Positivity of \(\rho \) gives the first operator’s invertibility and self-adjointness. The remaining assertions follow from multiplication of rectangular matrices.
Let \(C_rY=(V^\dagger A^\tau Y)_\tau \) and suppose that \(Q_r=\sum _\tau A^{\tau \dagger }VV^\dagger A^\tau \) is an orthogonal projection. Write \(\mathcal Q_rY=Q_rY\). For every prefix length \(N\),
The last identity requires \(\operatorname{tr}\rho \ne 0\). No positivity or injectivity of the prefix tensor is required for these identities.
The exact tail Gram form is \(\operatorname{tr}(Y^\dagger Q_rZ)\). A boundary map whose Gram operator vanishes on a vector also vanishes on that vector. Applying this fact to \(I-\mathcal Q_r\) proves the two support identities. The residual Gram factorization and the limiting right-multiplication formula give the remaining identities.
With \(G_{N,r}=\Gamma _{N,r}^\dagger \Gamma _{N,r}\), define
Whenever \(Q_r\) is the exact tail Gram matrix, \(F_{N,r}-R_\rho =G_{N,r}-K_{\infty ,r}\). The inverse equations for \(H_{N,r}\) hold under the hypotheses of the following theorem.
Substitute \(K_{\infty ,r}=R_\rho \mathcal Q_r\) and expand the complementary term.
Assume \(D,E{\gt}0\), \(\rho {\gt}0\), and that \(Q_r\) is an orthogonal projection equal to the exact tail Gram matrix. Let \(\delta _N=\| K_N^B-K_\rho \| \) and \(u=\| R_\rho ^{-1}\| \). If \(u\delta _N\leq a{\lt}1\), then \(F_{N,r}\) is invertible, \(H_{N,r}\) is self-adjoint, and
Without the small-error condition, \(\| \Gamma _{N,r}\| ^2\leq \| R_\rho \| +\delta _N\). These are intermediate estimates for [ 225 , Lemma commutation(ii) ] , rather than an assertion of the three-interval projection estimate.
The extended error is at most \(\delta _N\), so the Neumann estimate at the fixed invertible operator \(R_\rho \) bounds \(F_{N,r}^{-1}\). The exact support identity gives \(F_{N,r}\mathcal Q_r=\mathcal Q_rF_{N,r}=G_{N,r}\). Thus its inverse commutes with \(\mathcal Q_r\), proving self-adjointness and the supported inverse equation. Finally, \(\| G_{N,r}\| =\| \Gamma _{N,r}\| ^2\) and \(\| R_\rho \mathcal Q_r\| \leq \| R_\rho \| \) give the boundary-map bound.
Assume \(D{\gt}0\), and let the positive-dimensional prefix tensor \(B\) be primitive with faithful invariant matrix \(\rho \). Suppose that every exact tail Gram matrix \(Q_r\) is an orthogonal projection. For all sufficiently large prefix lengths, uniformly over every \(r\geq 0\), the preceding inverse and support equations hold and
Whenever \(u\delta _N\leq \tfrac 12\),
Consequently this inverse error tends to zero along every diverging prefix length, with arbitrary tail lengths. Under the cyclic hypotheses of Theorem 12.9.194, the exact projections are \(Q_r=P_{a+r}\); no restriction on the tail length is needed.
Primitive prefix convergence gives \(\delta _N\to 0\), so eventually \(u\delta _N\leq \tfrac 12\) and \(\delta _N\leq 1\). The resolvent identity bounds the inverse displacement by \(\| F_{N,r}^{-1}\| \, \| F_{N,r}-R_\rho \| \, u\). Multiplication by the contractive row projection preserves this bound. The estimates are independent of \(r\), so the same convergence holds for every choice of tails and every filter along which the prefix diverges.
Let \(H_N\) be the canonical range-\(R\) open Hamiltonian, and let \(H_L,H_T\) be its restrictions to a prefix of length \(n\) and a terminal interval of length \(W\), respectively. Then \(H_L+H_T\le 2H_N\). If \(n\le N\) and \(N+R\le n+W+1\), every interaction window lies in one of these intervals, and \(\ker H_L\cap \ker H_T=\ker H_N\). If the finite kernels equal the MPS boundary spaces and the overlap has length at least \(R\), the corresponding extended boundary spaces intersect exactly in \(G_N\).
Each interval contains a subset of the positive local terms of \(H_N\). This proves the energy upper bound. A vector annihilated by an interval sum is annihilated by every local projection in that sum. The covering condition therefore gives the kernel identity. The boundary-space identity follows by the physical coordinate isometries adjoining free sites.
For fixed \(R,W\) and \(m{\gt}0\), the finitely many canonical open Hamiltonians at lengths \(W+r\), \(0\le r{\lt}m\), have a common positive gap on their actual kernel complements. A gap valid at all lengths \(Km+Mm+r\) with \(K\ge K_0\) and \(r{\lt}m\) is valid at every \(N\ge (K_0+M)m\).
Take the minimum of the finitely many positive finite-dimensional gaps. For the second assertion use \(r=N\bmod m\) and \(K=\lfloor N/m\rfloor -M\).
Suppose \(m{\gt}0\), \(R{\gt}0\), \(Mm\ge R\), and the canonical open kernels equal \(G_N\) for every \(N\ge R\). Cover \(N=Km+Mm+r\) by the aligned prefix of length \((K+M)m\) and the terminal interval of length \(Mm+r\). Assume that, uniformly for \(K\ge K_0\) and \(0\le r{\lt}m\),
If the aligned prefix gaps have a common bound \(\gamma {\gt}0\), the full open Hamiltonians have a common bound \(\min (\gamma ,\eta )/16{\gt}0\), where \(\eta \) is a derived minimum over the terminal residues. For a normalized periodic tensor of period \(m\), the aligned gap is derived from its primitive blocked presentation. Consequently all \(N\ge (K_0+M)m\) have a common positive gap. The displayed projection estimate remains an explicit hypothesis; this criterion does not establish its periodic-tensor decay. It is the finite-Hamiltonian consequence of the three-interval estimate in [ 225 , Section 6, Lemma commutation(ii) ] .
Adjoining free sites preserves each interval gap. Their energies sum to at most twice the full energy, so the two-projection comparison gives the stated bound. The aligned periodic gap follows by blocking and positive comparison; the terminal gaps require only finite dimensionality. Quotient reconstruction then removes the residue restriction.
Let \(B\) have bond dimension \(E\) and satisfy \(\sum _i B_i^*B_i=I\). For every prefix length \(K\geq 0\),
Consequently, the map \(J_1\) of Definition 12.9.197 satisfies \(\| J_1\| \leq \sqrt E\), independently of \(K\). This is the prefix bound in the overlapping-window argument of [ 225 , Lemma 6.1(ii) ] .
Taking the trace of \(\sum _{|u|=K}(B^u)^*B^u=I\) gives the first identity. For a rectangular matrix \(Y\), Frobenius submultiplicativity then gives
Let \(T_i\) be the right residual window formed from \(A,B_i,V_i\), and let \(L_j\) be the left window formed from \(B_j\), in the coordinates of Definition 12.9.197. Their middle interval has length \(M\), the outer blocked prefix has length \(K\), and the original tail has length \(r\). For rectangular boundary fibers \(x_u\) and square boundary fibers \(y_\tau \),
The two sector bond dimensions may differ, and every length may be zero. If \(\varepsilon \geq 0\) and \(|\langle v,w\rangle |\leq \varepsilon \| v\| \| w\| \) for every \(v\in \mathcal G_M(B_i)\) and \(w\in \mathcal G_M(B_j)\), then
No normalization, injectivity, intertwining, or boundary inverse is assumed.
Fix \((u,\tau )\) and sum over the middle configuration. Trace cyclicity moves the blocked prefix word to the right boundary of the left window. Both resulting fibers lie in the stated middle ground spaces. The sums of their squared norms are exactly the squared norms of the full window vectors. The triangle inequality and finite Cauchy–Schwarz therefore preserve the coefficient \(\varepsilon \). The relative overlap estimate gives the projection-product bound.
Suppose \(B_i\) and \(B_j\) are inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices. For every \(\varepsilon {\gt}0\), all sufficiently large middle lengths \(M\) satisfy, for every \(K,r\geq 0\),
Thus the projection product tends to zero along every diverging middle length, while both outer lengths may vary arbitrarily. The original tensor \(A\) and support matrix \(V_i\) require no normalization or intertwining hypothesis for this conclusion. This is the cross-sector part of the projection comparison in [ 225 , Lemmas disjoint and commutation(ii) ] .
Primitive inequivalence makes the middle ground-space angle arbitrarily small. The preceding comparison transfers that bound uniformly over both outer lengths. Apply it with half the desired tolerance to obtain convergence along any filter with diverging middle length.
Let \(U,V,W\) and \(S_i,T_i,C_i\) be subspaces of a finite-dimensional real or complex Hilbert space, with \(i\) ranging over a finite set \(I\). Write \(P_Z\) for the orthogonal projection onto \(Z\), and set \(e_U=\| P_U-\sum _iP_{S_i}\| \), \(e_V=\| P_V-\sum _iP_{T_i}\| \), and \(e_W=\| \sum _iP_{C_i}-P_W\| \). Then
No orthogonality of the sector subspaces is required. This finite-sum estimate assembles the projection comparisons in [ 225 , Lemmas commutation(ii) and disjoint ] .
Put \(P=\sum _iP_{S_i}\), \(Q=\sum _iP_{T_i}\), and \(Z=\sum _iP_{C_i}\). Expand \(P_UP_V-P_W=(P_U-P)P_V+P(P_V-Q)+(PQ-Z)+(Z-P_W)\). Every orthogonal projection has norm at most one, so \(\| P\| \leq |I|\). Expand \(PQ-Z\) as the stated double sum and apply the triangle inequality.
Let the subspaces in the preceding theorem vary along an arbitrary filter, allowing the finite-dimensional ambient Hilbert spaces to vary. Suppose the three projection-sum errors tend to zero, and suppose \(\| P_{S_i}P_{T_i}-P_{C_i}\| \to 0\) for each \(i\), while \(\| P_{S_i}P_{T_j}\| \to 0\) for each \(i\ne j\). Then \(\| P_UP_V-P_W\| \to 0\). The finite index set is fixed and may be empty.
The finite double sum of sector errors tends to zero. Apply the finite sector projection estimate and squeeze the nonnegative joint defect between zero and its vanishing upper bound.
Let the virtual dimension \(E\) be positive and let \(\rho \in M_{E}(\mathbb {C})\) be positive definite. Put \(G_\infty =\operatorname {gramReshuffle}(\mathcal P_\rho )\) and \(R_\rho Y=Y\rho /\operatorname{tr}\rho \). For the correlated tail map \(C_r\),
Suppose the exact tail Gram matrix is an orthogonal projection \(Q\), so that \(C_r^*C_r=\mathcal Q\) with \(\mathcal QY=QY\). Set \(K_r=C_r^*G_\infty ^{\oplus \tau }C_r\) and \(I_r=\mathcal Q R_\rho ^{-1}\). With the residual window coordinates,
No transfer fixed-point equation, injectivity, or normalization is required for these identities.
The adjoint is \(C_r^*\mathbf y=\sum _\tau A^{\tau *}VY_\tau \). Right multiplication by \(\rho /\operatorname{tr}\rho \) commutes with this recombination, and \(R_\rho ^{-1}\) cancels it, giving (490). The projected tail Gram gives \(C_r\mathcal Q=C_r\) and hence \(\mathcal Q C_r^*=C_r^*\). Thus \(QZ=Z\) for \(Z=\sum _\tau A^{\tau *}VY_\tau \). Now \(J_1Z=(ZB^u)_u\) and \(K_r(ZB^u)=ZB^u\rho /\operatorname{tr}\rho \). Expand the resulting Frobenius pairing over \(\tau \) to obtain (491).
Under the hypotheses of Theorem 12.9.223, define \(D_M=T^*L_1-K_r^{\oplus u}J_1I_rC_r^*G_\infty ^{\oplus \tau }\). Then
This is an exact finite-volume identity. A quantitative bound or a limiting projection estimate requires further hypotheses on the prefix Gram.
Subtract the limiting mixed-Gram identity from the exact mixed-Gram pairing, and distribute subtraction through the two finite sums.
Decode a blocked configuration \(v\) of length \(M\) into its \(ML\) original sites and append an original tail \(\tau \) of length \(r\). This gives a bijection \(e_{M,r}\) from the pairs \((v,\tau )\) to the original configurations of length \(ML+r\), including zero lengths.
Let \(\Gamma _{B,V}(M,r)\) be the residual boundary map and let \(U_{M,r}\) be physical reindexing through \(e_{M,r}\). The right window map is the finite direct sum of \(U_{M,r}\Gamma _{B,V}(M,r)\) over the blocked prefix configurations \(u\) of length \(K\). No normalization or intertwining hypothesis is required.
Both maps have coefficient \(\operatorname{tr}(B^vV^*A^\tau X_u)\) at \((u,v,\tau )\).
Let \((B_j)_j\) be a finite family of pairwise gauge-phase inequivalent primitive tensors with faithful invariant densities, and let \(V_j\) be arbitrary rectangular embeddings. Write \(S_j(K,M,r)\) for the range of the right window map of sector \(j\). As \(M\to \infty \),
uniformly in arbitrary varying prefix lengths \(K\) and original-tail lengths \(r\). The assertion holds along any filter on which \(M\) tends to infinity. Residual-sector projection-sum decay also persists under any varying physical configuration isometry before adding the spectator. This is the right-window comparison in [ 225 , Lemma 6.1(i) ] .
Physical isometries preserve the relative angles between residual sectors. The resulting projection-sum error tends to zero by primitive sector separation. The right window acts independently on each prefix fiber, and for such fiberwise maps the error between the sum of the sector range projections and the joint range projection is at most the error of a single fiber. Hence adding the prefix spectator does not increase the operator-norm error.
Let \(X\) be a nonempty set, let \(\mathcal F\) be a filter on an index set \(I\), and let \(e:I\times X\to \mathbb R\). Suppose \(e(i,x_i)\to 0\) along \(\mathcal F\) for every function \(i\mapsto x_i\) from \(I\) to \(X\). Then, for every \(\varepsilon {\gt}0\), eventually \(e(i,x)\leq \varepsilon \) simultaneously for every \(x\in X\). No topology or countability assumption on \(X\) is required, and \(e\) need not be nonnegative. This elementary implication applies to the uniform projection estimates in [ 225 , Lemma commutation(ii) ] .
For each \(i\), select a parameter violating the upper bound if one exists; otherwise select any element of \(X\). Convergence along this selection gives \(e(i,x_i){\lt}\varepsilon \) eventually. The choice of \(x_i\) then implies \(e(i,x)\leq \varepsilon \) for every \(x\).
Let \(S,I,J\) be finite coordinate sets and let \(T\colon \mathbb {C}^I\to \mathbb {C}^J\). Write \(\mathcal F_S(T)\) for the operator applying \(T\) to every fiber of \(\mathbb {C}^{S\times I}\). Then
For any sector family \(T_j\colon \mathbb {C}^{I_j}\to \mathbb {C}^J\) with varying finite domains and common finite target, the joint-range projection satisfies
The sector index in this identity may be infinite, with each sum interpreted as the linear span. For a finite sector family, \(\mathcal F_S\) preserves operator sums and
Empty spectator sets and empty sector families are included. These finite-dimensional identities supply the spectator projection transport used in [ 225 , Lemma commutation(i) ] .
Choose a preimage independently in each fiber to characterize the range. The adjoint acts fiberwise, which gives the orthogonal-range criterion. The orthogonal projection is characterized by its range and orthogonal complement. For a joint span, orthogonality to every sector and membership of every fiber commute as universal quantifiers. Taking orthogonal complements again proves the joint-range identity. The final estimate follows from linearity of \(\mathcal F_S\) and its norm bound.
Let \(A\) be a periodic tensor of period \(m{\gt}0\) on a nonzero physical space. For \(K,M,r\geq 0\), let \(P\) denote the ground-space projection on \(Km+Mm+r\) sites, let \(P_{\mathrm L}\) project onto the prefix ground space on \(Km+Mm\) sites tensored with the free \(r\) sites, and let \(P_{\mathrm R}\) project onto the terminal ground space on \(Mm+r\) sites tensored with the free \(Km\) sites. Suppose that, for every pair of sequences \(K_M,r_M\), \(\| P-P_{\mathrm L}P_{\mathrm R}\| \to 0\) as \(M\to \infty \). Then there exist \(R{\gt}0\), \(N_0\geq R\), and \(\delta {\gt}0\) such that every \(N\geq N_0\) and every \(v\perp \ker H_{A,R,N}\) satisfy \(\delta \| v\| \leq \| H_{A,R,N}v\| \), where \(H_{A,R,N}\) is the canonical open parent Hamiltonian. At the same range, \(\ker H_{A,R,N}=G_A(N)\) for every \(N\geq R\). The convergence assumption is explicit; this is an auxiliary limit criterion for the projection comparison in [ 225 , Section 6, Lemma commutation(ii) ] .
Periodicity gives a range \(R\) for which every canonical open kernel at volume \(N\geq R\) is the original boundary space. Apply the selection uniformization to the parameter \((K,r)\) and tolerance \(1/2\). Choose one \(M\geq R\) for which all these defects are at most \(1/2\). Since \(m\geq 1\), the overlap \(Mm\) contains every interaction window. The residue cover criterion with prefix threshold zero supplies a common positive gap for every \(N\geq Mm\). Take \(N_0=Mm\).
For lengths \(K,M,r\geq 0\), the left-window coordinate bijection sends a pair \((\tau ,\sigma )\) of an original tail \(\tau \) of length \(r\) and a blocked configuration \(\sigma =u\mathbin {\frown }v\) of length \(K+M\) to \((u,(v,\tau ))\).
Let \(L_B(K,M,r)\) denote the left residual window of Definition 12.9.197. The coordinate bijection of Definition 12.9.231,
splits the blocked configuration into lengths \(K,M\) and moves the original tail to the last coordinate. If \(U_{K,M,r}\) is the associated unitary permutation, then
All lengths may be zero. No normalization or injectivity is assumed.
The coefficient of the left side is \(\operatorname{tr}(B^uB^v y_\tau )\). Concatenating the blocked words gives the same coefficient for the ordinary prefix boundary map applied to the tail fiber \(y_\tau \).
For every finite sector family \((B_j)_j\) with possibly different bond dimensions,
This comparison includes empty families and requires no normalization. Suppose in addition that the sectors are pairwise inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices. Then for every \(\varepsilon {\gt}0\), all sufficiently large \(M\) satisfy, for every \(K,r\geq 0\),
Consequently the left-window projection-sum error tends to zero along every diverging middle interval, with arbitrary outer-prefix and original-tail lengths. This is the aligned left-sector comparison for [ 225 , Lemma commutation(i) ] .
A unitary permutation transports orthogonal projections and preserves the operator norm. The finite-family projection identity then bounds the spectator projection-sum error by the ordinary prefix error. Primitive inequivalence makes that prefix error arbitrarily small at all sufficiently large lengths. Since \(K+M\geq M\), the threshold is uniform in \(K\), and the comparison is independent of \(r\).
The two maps to the common family of square virtual boundaries have fibers \(\mathcal A\mathbf x=(V^*A^\tau X_u)_{u,\tau }\) and \(\mathcal B\mathbf y=(Y_\tau B^u)_{u,\tau }\). They are finite direct sums followed by isometric coordinate rearrangements.
The fibers are the matrices in Definition 12.9.234, and \(\| \mathcal A\| \le \| C_r\| \), \(\| \mathcal B\| \le \| J_1\| \). Under the hypotheses of Theorem 12.9.223,
These are finite-dimensional estimates underlying the mixed-pairing step of Nachtergaele’s commutation lemma [ 225 ] ; no transfer normalization or injectivity is assumed here.
The coordinate rearrangements preserve norms, and finite direct sums do not increase operator norms. The fiber formulas identify the centered pairing with \(\mathcal A^*(G_B(M)-G_\infty )^{\oplus (u,\tau )}\mathcal B\). The norm bound follows from submultiplicativity.
Let \(B_j\) be a fixed finite family of pairwise inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices. Let \(A\) and the rectangular matrices \(V_j\) be arbitrary. For the full residual window maps \(C_j(K,M,r)\), if \(M\) tends to infinity along an arbitrary filter, then
The prefix \(K\) and tail \(r\) may vary arbitrarily. No normalization or intertwining is required for \(A\) or \(V_j\). This is the full-window sector sum in the projection comparison of [ 225 , Lemma commutation(i) ] .
The physical configuration isometry identifies \(C_j(K,M,r)\) with the residual boundary map at prefix length \(K+M\) and tail length \(r\). Since \(K+M\geq M\), the total prefix tends to infinity. Apply the isometric residual-sector projection-sum limit.
Suppose \(\sum _jV_jV_j^*=1\) and \(V_j^*A^{(L)}_i=(B_j)_iV_j^*\). Let \(U\) be the physical configuration isometry from the original \((K+M)L+r\) sites to the three-window coordinates. Then \(\sum _j\operatorname {ran}C_j(K,M,r)=U G_A((K+M)L+r)\). This includes zero lengths and requires no primitivity or injectivity.
Transport the exact residual-sector ground-space decomposition by \(U\). A linear map preserves sums of subspaces, and the range of a composite is the image of the original range.
Physical reindexing identifies the full window map \(C\) with the residual boundary map on \(K+M\) blocked sites and an original tail of length \(r\). Consequently,
This holds for all lengths, including zero, without injectivity or normalization hypotheses.
The configuration reindexing is an isometry. In \(C^*C\), its adjoint times itself is the identity, leaving the residual Gram.
For the right, left, and full window maps \(T,L_1,C\) and the two virtual factors \(J_1,C_r\), set
where \(H_{K+M,r}\) is the supported full-boundary Gram inverse.
Let \(D,E{\gt}0\), and let the blocked tensor \(B\) be primitive with a faithful invariant matrix \(\rho \). Suppose the correlated tail Gram is an orthogonal projection \(Q_r\) for every tail length \(r\). Along any filter with \(M\to \infty \), and with arbitrary \(K,r\),
This is the same-sector mixed-Gram estimate used in commutation (ii), equation (3.15), of [ 225 ] ; the algebraic row-support hypotheses are stated explicitly here.
The centered limiting mixed Gram is bounded by the common-prefix Gram error and hence tends to zero. Replace successively the right-window Gram, the supported full-boundary inverse, and the left-window Gram by their limits. Their errors tend to zero, since \(K+M\ge M\), and the remaining products have uniform bounds. Apply the three-operator perturbation estimate and then the triangle inequality.
For bounded linear operators \(A,a:G\to W\), \(B,b:F\to G\), and \(C,c:E\to F\) between normed spaces,
No finite-dimensional or completeness hypothesis is required.
Use \(ABC-abc=(A-a)BC+a(B-b)C+ab(C-c)\), the triangle inequality, and submultiplicativity of the operator norm.
Let the normed spaces and operators in Lemma 12.9.241 vary along an arbitrary filter. Suppose the three individual differences tend to zero in norm, and the products \(\| B\| \| C\| \), \(\| a\| \| C\| \), and \(\| a\| \| b\| \) have a common eventual upper bound. Then \(\| ABC-abc\| \to 0\).
Bound the product difference by the common bound times the sum of the three individual norm differences, and apply the squeeze theorem.
Decode the blocked prefix \(u\) of length \(K\) and blocked middle \(v\) of length \(M\) into configurations of lengths \(KL\) and \(ML\). Concatenation with the original tail \(\tau \) gives a configuration on \(KL+ML+r\) sites. This bijection is the full residual-window configuration bijection followed by the identification \((K+M)L=KL+ML\), and it sends \((u,v,\tau )\) to the concatenation of the three configurations in that order. All zero lengths are included. These are the physical covering coordinates of [ 225 , Section 6 ] .
Let \(A\), \(B_j\), and \(V_j\) satisfy \(\sum _jV_jV_j^*=1\) and \(V_j^*A^{(L)}_i=(B_j)_iV_j^*\). Let \(U\) be the configuration isometry from the original \(KL+ML+r\) sites to the physical cover coordinates. Then
Here the terminal boundary map leaves the first \(KL\) sites free and places the original ground space on the last \(ML+r\) sites. No primitivity, injectivity, or isometry of the virtual embeddings is required. This is the right joint-window identification in the projection comparison of [ 225 , Lemma commutation(ii) ] .
Each right residual window is the direct sum, over prefix configurations, of its physically reindexed residual boundary map. The exact residual-sector decomposition identifies the joint fiber range with the original ground space on \(ML+r\) sites. Taking finite direct sums preserves the joint range. In the physical cover coordinates, the terminal boundary map is the same direct sum of ordinary boundary maps, after a bijection of its prefix boundary coordinates. That bijection is surjective, so it does not change the range.
Let \(A\) be an MPS tensor and let \(U_{K,M,r}\) be the unitary from original configurations to the three-window coordinates of Definition 12.9.197. Then
Suppose a family \((B_j)_j\) presents the blocked prefix ground space exactly:
Then its joint left-window range is the original aligned prefix ground space, with the tail unconstrained:
In particular, this equality follows for a finite family with an isometric virtual resolution satisfying
All lengths may be zero. No primitivity, normalization, or injectivity is assumed. This is the exact left-window identification used in the three-interval comparison of [ 225 , Lemma commutation(ii) ] .
Concatenating the two blocked words and decoding their physical letters identifies the blocked left map with the original prefix boundary map. Each tail configuration contributes an independent copy of the prefix space. The joint span of these finite families is therefore obtained by applying the joint prefix-space projection independently on each tail fiber. The isometric virtual resolution supplies the stated prefix presentation by trace cyclicity.
Under the virtual resolution and blocked cointertwining hypotheses of Theorem 12.9.237, let \(U\) be the configuration isometry from the original \(KL+ML+r\) sites to the physical cover coordinates. Then \(\sum _j\operatorname {ran}C_j(K,M,r)=U G_A(KL+ML+r)\). The statement includes zero lengths and requires no primitivity or injectivity.
Apply the exact full-window joint-range identity and transport it through \((K+M)L+r=KL+ML+r\). The cover configuration bijection is precisely the full-window bijection followed by this arithmetic identification.
Let \(U,V,W\) be closed subspaces of a Hilbert space and let \(F\) be a surjective linear isometry to another Hilbert space. Then
Orthogonal projections are conjugated by \(F\), and conjugation by a surjective isometry preserves the operator norm. Taking the adjoint reverses the order of the two projections and preserves the norm.
Under the isometric virtual resolution and blocked letter intertwining of Theorem 12.9.245, let \(U^{\mathrm{cover}}_{K,M,r}\) be the unitary from original configurations of length \(KL+ML+r\) to the three-window coordinates of Definition 12.9.243. Then
All lengths may be zero. This is the same physical coordinate isometry used for the right and full windows; no additional tensor hypothesis is imposed. The equality supplies the aligned prefix space in [ 225 , Lemma commutation(ii) ] .
Transport the previous left-window equality along the arithmetic identification \((K+M)L=KL+ML\). By its definition, the cover bijection is exactly the previous configuration bijection followed by this identification.
Let \(T,L_1,C\) denote the right, left, and full window maps. Suppose that \(Q_r\) is the exact correlated-tail Gram projection, that the right and full supported Gram inverses satisfy \(G_{M,r}H_{M,r}=Q_r\) and \(G_{K+M,r}H_{K+M,r}=Q_r\), with \(H_{M,r}\) self-adjoint, and that the ordinary Gram \(G_B(K+M)\) is invertible. Then
Here the hats denote the direct sums over the corresponding exterior configurations. This is the supported-boundary cancellation in commutation (ii), equation (3.15), of [ 225 ] .
The right Gram and its supported inverse act independently on prefix configurations. The left Gram and its inverse act independently on tail configurations. Both factorizations of the full boundary map have the exact row support \(Q_r\). Substitute these identities into the supported range-projection cancellation formula.
Let \(D,E{\gt}0\), and let \(B\) be primitive with a faithful invariant matrix \(\rho \). Suppose the exact correlated-tail Gram is an orthogonal projection \(Q_r\) for every \(r\). Along any filter with \(M\to \infty \) and arbitrary exterior lengths \(K,r\),
This is the same-sector part of commutation (ii) in [ 225 ] ; the tail support is retained explicitly.
The primitive supported-inverse estimates supply the inverse identities and uniform bounds at lengths \(M\) and \(K+M\). The ordinary primitive Gram inverse is uniformly bounded at length \(K+M\). The window maps are uniformly bounded by their boundary-map norms. Apply the norm estimate for the supported cancellation formula to the vanishing mixed residual.
Let \(D{\gt}0\), and let \((B_j)_{j\in J}\) be a finite family of positive-dimensional primitive tensors with faithful invariant matrices, pairwise inequivalent up to similarity and phase. Let \(T_j,L_j,C_j\) be the right, left, and full window maps arising from an original tensor and rectangular embeddings \(V_j\). Suppose that each correlated-tail Gram is an orthogonal projection at every tail length. Write \(\mathcal T=\sum _j\operatorname {ran}T_j\), \(\mathcal L=\sum _j\operatorname {ran}L_j\) and \(\mathcal C=\sum _j\operatorname {ran}C_j\). Along any filter with \(M\to \infty \) and arbitrary exterior lengths \(K,r\),
Replace each joint projection by the sum of its sector projections. The three replacement errors vanish. The paired sector defects vanish by the supported same-sector cancellation, and the different-sector projection products vanish by primitive inequivalence. The finite-sector projection estimate combines these limits.
Let \(A\) be a normalized periodic tensor of period \(m\). There exist positive-dimensional blocked sectors \(B_j\), indexed by \(j\in \mathbb Z/m\mathbb Z\), and rectangular isometries \(V_j\) such that
and, for every pair of sequences of exterior lengths \(K(M),r(M)\),
The spaces are the joint residual window ranges in the triple configuration coordinates. This establishes their projection comparison from periodicity alone; identifying them with original physical interval spaces is a separate isometric transport. The estimate is the residual-coordinate form of commutation (ii) in [ 225 ] .
Choose the faithful primitive, inequivalent cyclic sector resolution. Its cyclic shift is \(P_{j+1}A_i=A_iP_j\). Reversing the labels permits the word shift formula to show that the correlated-tail Gram is precisely \(P_{j-r}\), an orthogonal projection. Apply joint residual window projection decay and retain the two intertwiner identities of the same resolution.
Let \(A\) be a normalized periodic tensor of period \(m\). For arbitrary sequences \(K_M,r_M\in \mathbb N\), put \(N_M=K_Mm+Mm+r_M\). Let \(P_{\mathrm L,M}\) and \(P_{\mathrm R,M}\) be the ground-space projections for the prefix of length \((K_M+M)m\) and the suffix of length \(Mm+r_M\), extended to the full interval. Then
This is the explicit periodic tensor form of the cover estimate used in [ 225 , Lemma commutation (ii) and Section 6 ] . The passage from a general GVBS presentation to this tensor presentation remains separate; see [ 61 ] .
The cyclic primitive sector resolution gives the joint three-window projection limit. The right, left, and full joint ranges are the images of the corresponding original interval spaces under one physical isometry. Isometric conjugation preserves the norm; taking adjoints reverses the projection order and gives the displayed defect.
Let \(A\) be a normalized periodic tensor. There exist \(R\ge 1\) and \(\delta {\gt}0\) such that the canonical range-\(R\) open-parent Hamiltonian \(H_{R,N}\) satisfies
There is no divisibility condition on \(N\). The interaction range and constant are derived from periodicity alone. This is the normalized periodic tensor specialization of [ 225 , Theorem 1.2 and Section 6 ] ; the general GVBS presentation’s identification with these support spaces remains separate, as recorded in [ 61 ] .
The physical projection limit and the selection argument give one common cover length and a positive gap at every sufficiently large volume. The original cyclic intersection gives the exact kernels at the same interaction range. Each of the remaining finitely many volumes has a positive finite-dimensional norm gap on its actual kernel complement; taking a finite minimum completes the uniform bound.
For every normalized periodic tensor \(A\), there exist \(R\ge 1\), a positive local matrix \(h\) on \(R\) sites, and \(\delta {\gt}0\) such that its open-chain interaction sums \(H_N\) satisfy
No local interaction, invariant density, or sector separation is supplied. The scope is the explicit normalized periodic tensor setting of [ 225 , Theorem 1.2, Lemma existenceinteraction, and Section 6 ] .
Choose the configuration-basis matrix of the canonical local ground-space complement projection. It is positive, and its open interaction sum is exactly the canonical parent Hamiltonian at every volume.
Let \(A\) be a normalized periodic tensor. There exist a range \(R\ge 1\), a positive local matrix \(h\) on \(R\) sites, and constants \(\delta ,\gamma {\gt}0\) with the following properties. The open interaction sum \(H_N\) satisfies
For every pure zero-energy state \(\varphi \) of this same interaction and every observable \(X\) supported on \([a,a+k)\) with \(k{\gt}0\) and \(\varphi (X)=0\), along every filter for which both margins \(\ell _n,r_n\to \infty \), there is a limit
Both constants precede every state, observable and interval exhaustion. The interaction, its range and all support data are derived from periodicity. This is an existence consequence of Theorems 1.1–1.2 of [ 225 ] in the explicit normalized periodic tensor setting. The general GVBS presentation remains separate; see [ 61 ] .
Choose the positive canonical interaction and its range from the all-volume finite-chain gap theorem. Its exact kernel identities imply the eventual kernel hypothesis of the periodic infinite-volume theorem. Apply that theorem to the same interaction, and retain the finite-chain constant and the common pure-state commutator constant.
Let \(A\) be an MPS tensor of positive bond dimension, let \(\rho {\gt}0\), and let \(X\) act on \(k\geq 0\) sites. Define \(\eta _A\) by the trace insertion formula, without assuming transfer invariance of \(\rho \). Then
Tensor normalization, injectivity, and primitivity are unnecessary. This is the finite support characterization used for the faithful generators in [ 225 , Section 3 and the proof of the overlap lemma in Section 4 ] ; the construction of minimal faithful generators from a general GVBS presentation remains separate [ 61 ] .
Put \(C_b=\sum _\sigma X_{b\sigma }A^\sigma \). Expansion and cyclicity of the trace give
The sum is positive. Faithfulness of \(\rho \) implies that the final trace vanishes precisely when every \(C_b\) vanishes. Since \((X\Gamma _k(Y))_b=\operatorname {tr}(C_bY)\) for every boundary matrix \(Y\), nondegeneracy of the trace pairing is equivalent to \(X\Gamma _k=0\). The range of \(\Gamma _k\) is \(G_A(k)\).
Let \(A\) and \(B\) be normalized periodic tensors. Suppose that there is \(z\ne 0\) such that their periodic vectors satisfy \(|V^{(N)}(B)\rangle =z^N|V^{(N)}(A)\rangle \) for every \(N{\gt}0\). Then their bond dimensions agree, they are gauge and phase equivalent, and \(\mathcal G_A(N)=\mathcal G_B(N)\) for every \(N\ge 0\).
Along common multiples of the two periods, the self-overlaps tend to the positive periods. Proportionality gives the exact identity \(|\langle V^{(N)}(A) | V^{(N)}(B) \rangle |^2 =\langle V^{(N)}(A) | V^{(N)}(A) \rangle \langle V^{(N)}(B) | V^{(N)}(B) \rangle \). Thus the mixed overlap cannot tend to zero. The periodic overlap dichotomy supplies the gauge relation. Gauge transformations and nonzero scalar rescaling preserve all open-boundary spaces.
Let \((A_j)_{j=0}^{b-1}\) be a finite family of normalized periodic tensors, with arbitrary periods, and let \(\mu _j\ne 0\). There is a subfamily selected by an injective map \(s:\{ 0,\ldots ,g-1\} \to \{ 0,\ldots ,b-1\} \) such that its members are pairwise inequivalent under gauge and phase transformations. Every original member is gauge and phase equivalent to a selected member, after the corresponding bond-dimension identification.
The selected family inherits the original periods and positive bond dimensions. The statement also allows an empty original family. This is an algebraic support identity; it does not assert the joint intersection property or automatic existence of a parent interaction.
Select one member from each periodic-vector phase class. The preceding lemma identifies the open-boundary spaces of all members of a class. Taking their sums proves the displayed identity. The phase-class separation gives pairwise gauge and phase inequivalence.
Let \(A\) be trace preserving, let \(V^*V=I\), and suppose \(A_iV=VB_i\) for every letter \(i\). Then \(B\) is trace preserving. This is the normalization identity used in the compressed periodic sectors of [ 191 , Section 2.1 ] .
Taking products of adjoints in \(A_iV=VB_i\) and summing gives
Let \(A\) have positive bond dimension, assume \(\sum _iA_i^*A_i=I\), and let \(\rho {\gt}0\) satisfy \(E_A(\rho )=\rho \). There exist a nonempty finite family of positive-dimensional periodic tensors \(B_j\) with periods \(m_j\geq 1\) and support isometries \(V_j\) such that
For every \(N\geq 0\),
This is a consequence for supplied normalized faithful generating data in [ 225 , Section 3 ] . It does not construct such data from a general GVBS boundary limit or impose minimality; that passage remains separate [ 61 ] .
Faithful stationarity makes every invariant projection reducing. Decompose into irreducible isometric corners; the identity resolution and positive ambient bond dimension imply that the family is nonempty. Trace preservation passes to each corner. Each normalized irreducible corner therefore has a finite period. The letterwise intertwining identifies the MPS boundary spaces at every length, and the unit-weight direct sum has their joint space.
Let \(A\) have positive bond dimension and satisfy \(\sum _i(A^i)^*A^i=I\), and let \(\rho {\gt}0\) be a virtual matrix. For every \(k\geq 0\), there is a density matrix \(\sigma _k\) such that
The trace representation holds for every observable \(X\) on \(k\) sites. No transfer invariance of \(\rho \), injectivity, or primitivity is assumed. These finite densities identify the supports for supplied faithful generating data as in [ 225 , Section 3 and the overlap proof in Section 4 ] . The construction of faithful minimal data from a general GVBS presentation remains separate [ 61 ] .
The physical alphabet is nonempty by the positive physical dimension of a left-canonical tensor on a nonzero bond space. Trace preservation gives \(\operatorname {tr}E^k(\rho )=\operatorname {tr}\rho \), so \(\eta _A(I_k)=1\). Positivity and complex linearity therefore yield a representing positive density matrix of trace one. For positive \(\sigma _k\), zero trace pairing with \(X^*X\) is equivalent to \((X^*X)\sigma _k=0\). Since \(\ker (X^*X)=\ker X\), this is equivalent to \(\operatorname {ran}\sigma _k\subseteq \ker X\). The faithful support criterion identifies the same condition with \(G_A(k)\subseteq \ker X\). Testing the two orthogonal-complement projections gives equality of the two subspaces.
Let \((V_i)_{i\in I}\) be finitely many subspaces of a complex Hilbert space, and let \(B\) be a real matrix with zero diagonal and \(\| B\| {\lt}1\). Suppose that, for \(i\ne j\), \(x\in V_i\), and \(y\in V_j\),
Then the family \((V_i)\) is independent: a vanishing sum of vectors, one from each subspace, has every component zero.
The lower Gram estimate gives
A zero sum therefore makes every summand zero.
Let \(V_i(n)\) be a finite family of subspaces of Hilbert spaces that may depend on \(n\). Suppose that for every \(\varepsilon {\gt}0\), eventually
Then the family \((V_i(n))_i\) is eventually independent.
Let \(C\) have zero diagonal and all off-diagonal entries equal to one. Choose \(\varepsilon =(1+\| C\| )^{-1}\). Then \(\| \varepsilon C\| {\lt}1\), and the preceding lemma applies whenever the stated pairwise estimate holds.
A finite family of pairwise gauge and phase inequivalent primitive tensors, with faithful invariant matrices, has independent boundary spaces \(G_{B_i}(N)\) for every sufficiently large \(N\). This is the consequence of the overlap estimates in [ 225 , Lemmas disjoint and overlapestimate ] .
Inequivalence gives uniformly vanishing pairwise ground-space overlaps. Finiteness makes the bounds simultaneous, so the preceding theorem proves independence.
Let \((W_i)_{i\in I}\) be an independent family of subspaces, with \(I\) finite, and let \(U_i,V_i\subseteq W_i\). Then
More generally, suppose a family of linear restrictions \(f_t:E\to F\) determines every vector of \(E\). For any independent family \((G_i)\) in \(F\), the subspaces \(\bigcap _t f_t^{-1}(G_i)\) are independent in \(E\). In particular, adjoining the first and last physical indices preserves independence of middle support spaces. Independence is also unchanged by passing from the coefficient functions to their Euclidean realization.
Independence makes the decomposition into the ambient subspaces unique, so the decompositions from the two sums agree term by term. For the restriction assertion, apply independence after every restriction, then use that the restrictions determine the original vector. The first and last physical restrictions determine all coefficients. The Euclidean realization is a linear isomorphism.
Let \(G_i\subseteq (\mathbb C^d)^{\otimes n}\) and \(H_i\subseteq (\mathbb C^d)^{\otimes (n+1)}\) be finite families. Assume that the \(G_i\) are independent and that every restriction of a vector in \(H_i\), obtained by fixing either the first or last physical index, lies in \(G_i\). Write \(L_i=H_i\otimes \mathbb C^d\) and \(R_i=\mathbb C^d\otimes H_i\). Then
This is the geometric step of Nachtergaele, arXiv:cond-mat/9410110, the proposition and equivalence lemma of Section 4; no tensor representation is assumed in this statement.
Both \(L_i\) and \(R_i\) lie in \(\mathbb C^d\otimes G_i\otimes \mathbb C^d\). These ambient spaces are independent by the preceding lemma. Finite sums commute with adjoining an exterior physical index, as is seen by decomposing each fixed-index coefficient separately. Apply the independent ambient-sum intersection identity.
Let \((A_i)_{i\in I}\) be a finite family of tensors, possibly empty. Suppose that their ground spaces \(\mathcal G_i(n)\) are independent and that each tensor satisfies
Then the sums \(\mathcal G(q)=\sum _i\mathcal G_i(q)\) satisfy
The individual intersection identities and middle independence remain explicit hypotheses. This criterion does not by itself assert the unrestricted intersection theorem for periodic tensor families.
Ground-space vectors restrict into the corresponding ground spaces on the shorter interval. Apply the preceding geometric lemma, then substitute the assumed individual intersection identities.
Let \(A\) have positive bond dimension, assume \(\sum _iA_i^*A_i=I\), and let \(\rho {\gt}0\) satisfy \(E_A(\rho )=\rho \). Write \(\omega _{A,\rho }\) for the resulting quasi-local state. For every \(k\geq 0\), there exists a density matrix \(\sigma _k\geq 0\), of trace one, such that
Thus the same density describes every translated interval of length \(k\), and its support is exactly the tensor boundary space. The generating data are supplied, as in [ 225 , Section 3 and the overlap proof in Section 4 ] ; their construction from a general GVBS presentation remains separate [ 61 ] .
The quasi-local state agrees on each consecutive interval with the trace insertion functional. The finite-density support theorem therefore gives the same matrix \(\sigma _k\) at every position.
A tensor of positive bond dimension satisfying \(\sum _iA_i^*A_i=I\) has positive physical dimension. This is the left-canonical normalization of [ 191 , Section 2.1 ] .
An empty physical alphabet would make the sum zero, contradicting the nonzero identity on the bond space.
Let \(A\) have positive bond dimension, assume \(\sum _iA_i^*A_i=I\), and let \(\rho {\gt}0\) satisfy \(E_A(\rho )=\rho \). Let \(h\geq 0\) have range \(R{\gt}0\), and suppose that \(\ker H_h(N)=G_A(N)\) for every sufficiently large \(N\). There is one \(\gamma {\gt}0\) with the following property. For every pure zero-energy state \(\varphi \), every observable \(X\) supported on a positive-length interval \([a,a+k)\) with \(\varphi (X)=0\), and every filter along which \(\ell _n,r_n\to \infty \), put \(\Lambda _n=[a-\ell _n,a+k+r_n)\). Then the literal finite-volume limit exists and satisfies
Neither irreducibility nor periodicity of \(A\), nor translation invariance of \(\varphi \), is assumed. This is a consequence for supplied faithful normalized generating data in the setting of [ 225 , Theorem 1.2 and Sections 3, 6 ] . Constructing such data from an arbitrary GVBS boundary limit remains separate; see [ 61 ] .
Normalization implies positive physical dimension. The faithful stationary matrix yields a finite periodic family \(B_j\) with \(G_A(N)=G_{\bigoplus _jB_j}(N)\) at every length. Hence the same interaction has these joint spaces as its eventual kernels. Applying the periodic-family gap theorem with unit weights proves (535), with its constant chosen before the state, observable, or exhaustion.
Let \(A\) satisfy \(\sum _i(A^i)^*A^i=I\), let \(\rho \geq 0\) be a fixed point of its transfer map with \(\operatorname {tr}\rho \neq 0\), and let \(h\geq 0\) be an interaction of positive range whose open-chain kernels equal the MPS boundary spaces \(G_N(A)\) for all sufficiently large \(N\). Then the quasi-local state generated by \(A\) and \(\rho \) has zero local energy for \(h\). No primitivity, faithfulness of \(\rho \), or purity is assumed [ 225 , Theorems 1.1–1.2 ] .
The restriction of the generated state to a long interval is supported in \(G_N(A)\), which is the kernel of the open Hamiltonian there; hence every local interaction term has zero expectation.
Let \(A\) have positive bond dimension and satisfy \(\sum _i(A^i)^*A^i=I\), with a positive definite fixed point \(\rho \) of its transfer map, and let \(h\geq 0\) be an interaction of positive range whose open-chain kernels eventually equal \(G_N(A)\). The state generated by \(A\) and \(\rho \) has zero local energy, and there are a blocking length \(L{\gt}0\) and a nonempty finite family of pairwise inequivalent normalized primitive tensors \(B_j\) with positive definite fixed points such that the blocked tensor of \(A\) and the unit-weight direct sum of the \(B_j\) have the same boundary spaces at every length. The zero-energy states of \(h\) are exactly the convex combinations of the states generated by the \(B_j\), transported back through the blocking, and the pure zero-energy states are exactly these transported states [ 225 , Theorems 1.1–1.2 ] .
The faithful fixed point gives a periodic presentation of the boundary spaces; blocking by a common multiple of the periods yields primitive sectors with faithful fixed points. The classification of the zero-energy face for primitive sectors, transported through the blocking isomorphism of quasi-local algebras, gives both assertions.
Let \(T:E\to F\) be a complex linear map from a normed space to an inner-product space, and let \((W_i)_{i\in I}\) be a finite family of subspaces of \(F\). Suppose \(C_i\geq 0\) and
Suppose also that each \(x_i\in W_i\) admits a preimage \(u_i\) with \(Tu_i=x_i\) and \(\| u_i\| \leq C_i\| x_i\| \). Every \(x\in \sum _iW_i\) then admits a preimage \(u\) satisfying
This is a consequence of the supplied lower Gram bound, of the form used in the overlap estimate in [ 225 , Section 4 ] .
Choose a decomposition \(x=\sum _i x_i\) with \(x_i\in W_i\). The lower Gram bound gives \(\| x_i\| \leq 2\| x\| \) for every \(i\). Choose the stated preimages and put \(u=\sum _i u_i\). Linearity and the triangle inequality give the conclusion.
Let \(d\ge 1\) and let \(A\) be any tensor. Suppose that, for every \(q\ge R\),
If \(R{\gt}0\), the canonical open Hamiltonian of range \(R\) has kernel \(\mathcal G_A(N)\) for every \(N\ge R\). Consequently, if the one-step identity holds at every sufficiently large length, there is \(R_0{\gt}0\) such that the same kernel identity holds for every \(N\ge R\ge R_0\). The intersection identity is an explicit hypothesis of this criterion.
A vector in the Hamiltonian kernel satisfies every contiguous local ground-space constraint. Iterating the one-step intersections enlarges these local spaces to the whole interval. Conversely, every open-boundary vector satisfies every local constraint.
For a finite tensor family, eventual independence of the middle ground spaces and the individual eventual one-step intersection identities imply the eventual joint one-step intersection identity. In particular, this applies to any finite family of pairwise gauge and phase inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices. Thus, writing \(\mathcal G(q)=\sum _j\mathcal G_j(q)\), for all sufficiently large \(n\),
The family may be empty. The primitive assertion concerns the supplied generating tensors, rather than an arbitrary GVBS presentation.
Finiteness gives one eventual set on which all individual identities hold. Apply the middle-independence criterion on its intersection with the eventual independence set. For primitive tensors, overlap decay supplies that independence, and period-one periodicity supplies the individual intersections.
Let \(d\ge 1\), let \((A_j)\) be a finite family of pairwise gauge and phase inequivalent normalized primitive tensors with positive bond dimensions and faithful invariant matrices, and let \(\mu _j\ne 0\). For \(T=\bigoplus _j\mu _jA_j\), there exists \(R_0{\gt}0\) such that
In particular, the canonical local projection of range \(R_0\) is a positive parent interaction with this exact kernel at every \(N\ge R_0\). No simultaneous word span is supplied in this assertion or used in the family intersection argument.
Nonzero weights preserve the joint open-boundary space. Substitute the eventual joint intersection into the canonical kernel criterion. The orthogonal projection onto the local ground-space complement is positive, so choosing that projection gives the interaction.
Suppose \(A_iV=VB_i\) for every letter \(i\). For every virtual matrix \(\sigma \) and every observable \(X\) on \(k\geq 0\) sites,
If additionally \(V^*V=I\), \(\rho =V\sigma V^*\), and \(\operatorname {tr}\sigma \neq 0\), then \(\operatorname {tr}\rho =\operatorname {tr}\sigma \) and \(\eta _{A,\rho }(X)=\eta _{B,\sigma }(X)\). No normalization, stationarity, or positivity is needed for these identities. This concerns the finite insertion formula in [ 225 , Section 3 ] .
Letter intertwining gives \(A^\tau V=VB^\tau \) for every word \(\tau \). Expanding the insertion as a sum over words proves (541). When \(V^*V=I\), cyclicity gives \(\operatorname {Tr}(VZV^*)=\operatorname {Tr}Z\); apply this to both the inserted matrix and the normalization denominator.
Let \(A\) satisfy \(\sum _iA_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace with \(E_A(\rho )=\rho \). There exist a positive dimension \(E\), an isometry \(V\), a tensor \(B\), and a faithful matrix \(\sigma {\gt}0\) such that
\(\eta _{A,\rho }(X)=\eta _{B,\sigma }(X)\) for every observable on every finite interval, including length zero. All conclusions except \(\sum _iB_i^*B_i=I\) hold without the normalization of \(A\). This is a restriction of supplied stationary generating data to their support, as in [ 244 ] , restriction to full-rank fixed points, and the normalized generating representation of [ 225 , Section 3 ] . Constructing generating data from an arbitrary GVBS boundary limit remains separate [ 61 ] .
The stationary support projection \(P\) obeys \((I-P)A_iP=0\). Choose \(V\) with \(V^*V=I\) and \(VV^*=P\). Then \(PV=V\) gives \(A_iV=V(V^*A_iV)\). Compression preserves trace preservation and the stationary equation. Since \(P\rho P=\rho \), the compressed matrix is positive definite and expands as \(\rho =V\sigma V^*\). Cyclicity gives \(\operatorname {Tr}\sigma =\operatorname {Tr}\rho \), whose nonvanishing forces \(E{\gt}0\). Apply the insertion intertwiner to obtain all finite expectation identities.
For an MPS tensor \(A\), a projection \(P\), and \(L\geq 0\), let \(G_A^P(L)\) be the set of open-boundary vectors \(\Gamma _L^A(Y)\) with \(PYP=Y\). For \(P=I\) it is \(G_A(L)\).
Let \(V^*V=I\) and \(A_iV=VB_i\) for every \(i\). Then for every \(L\geq 0\),
The intertwining gives \(\Gamma _L^A(VXV^*)=\Gamma _L^B(X)\), and \(VXV^*\) lies in the corner of \(VV^*\). Conversely, if \(Y\) lies in that corner, then \(Y=V(V^*YV)V^*\).
Let \(A\) be an MPS tensor, let \(\rho \geq 0\) have nonzero trace and satisfy \(\sum _iA_i\rho A_i^*=\rho \), and let \(P\) be its support projection. The matrix \(\rho \) need not be faithful. For every \(k\geq 0\) and every \(k\)-site matrix \(X\),
If moreover \(\sum _iA_i^*A_i=I\), there is a density matrix \(\sigma _k\) with \(\eta _{A,\rho }(X)=\operatorname {tr}(\sigma _kX)\) for every \(X\) and \(\operatorname {ran}\sigma _k=G_A^P(k)\). This describes the local support spaces of normalized stationary generating data, as in [ 225 , Section 3 and the proof of the overlap lemma in Section 4 ] , without the faithfulness that the source obtains from minimality. Passing from an arbitrary GVBS boundary limit to normalized stationary data remains separate [ 61 ] .
Compress to the support of \(\rho \). The compressed data are faithful and have the same finite expectations, so the faithful support theorem and the finite-density theorem apply to them. The boundary space of the compressed tensor is \(G_A^P(k)\) by (542).
For a rectangular map \(V:\mathbb C^E\to \mathbb C^D\), let \(J_V(Y)=YV^*\) on the Hilbert space of rectangular matrices, with the Hilbert–Schmidt norm. Suppose that \(V^*(A^{[L]})^i=B^iV^*\) for every blocked letter \(i\). Then the residual boundary map satisfies \(\mathcal F^{N,r}_{A,B,V}=\Gamma ^A_{NL+r}J_V\) for every \(N,r\geq 0\). In particular, the boundary trace uses \(YV^*\) in the original site order.
The rectangular intertwiner moves \(V^*\) through each blocked letter. Thus the residual coefficient is \(\operatorname {tr}(A^{\widetilde u}A^\tau YV^*)\), which is the original boundary coefficient at \(YV^*\).
Let \(A\) be a normalized irreducible periodic tensor. There are \(C{\gt}0\) and \(n_0\) such that every \(n\geq n_0\) and \(x\in \mathcal G_n(A)\) have a virtual boundary \(u\) with \(\Gamma ^A_n(u)=x\) and \(\| u\| \leq C\| x\| \). The bound includes every residue length. This is the boundary normalization step in the mixed-transfer argument of [ 225 , Lemma disjoint ] , applied to periodic generating tensors using [ 191 , Lemma 6 ] .
Write \(n=Nm+r\) using the tensor’s period \(m\). Its primitive cyclic sectors give residual maps \(S_j\) and supported inverses \(H_j\) with \(S_jH_jS_j^*=P_{\operatorname {ran}S_j}\) and uniformly bounded \(\| H_j\| \| S_j\| \), independently of \(r\). The original lift of a sector vector \(x_j\) is \(J_{V_j}H_jS_j^*x_j\). Small pairwise residual overlaps give \(\tfrac 12\sum _j\| x_j\| ^2\leq \| \sum _jx_j\| ^2\). The finite sum of these bounded lifts therefore has norm at most \(2(\sum _jc_j)\| x\| \), where \(c_j\) bounds \(\| J_{V_j}\| \| H_j\| \| S_j\| \). Enlarging this constant makes it positive. Since \(N=\lfloor n/m\rfloor \) tends to infinity, the estimate holds at every sufficiently large \(n\).
Let \(A,B\) be normalized irreducible periodic tensors, possibly of different periods and bond dimensions. Suppose that whenever their bond dimensions agree, they are inequivalent up to similarity and phase. For every \(\varepsilon {\gt}0\), all sufficiently large \(n\) satisfy \(|\langle x,y\rangle |\leq \varepsilon \| x\| \| y\| \) for every \(x\in \mathcal G_n(A)\) and \(y\in \mathcal G_n(B)\). The statement concerns boundary spaces of generating tensors and does not assert purity of their associated states.
Irreducibility, normalization and inequivalence give a strict spectral radius bound for the original mixed transfer operator. Consequently \(\| \Gamma ^A_n{}^*\Gamma ^B_n\| \to 0\). Choose boundary lifts \(u,v\) with \(\| u\| \leq C_A\| x\| \) and \(\| v\| \leq C_B\| y\| \). Then \(|\langle x,y\rangle | \leq C_AC_B\| \Gamma ^A_n{}^*\Gamma ^B_n\| \| x\| \| y\| \).
A finite family of pairwise gauge and phase inequivalent normalized irreducible periodic tensors has independent original-chain boundary spaces at every sufficiently large length. No common period is prescribed.
Finitely many pairwise overlap bounds hold simultaneously at all sufficiently large lengths. Apply the finite lower Gram criterion for independence.
Let \(d\ge 1\), let \((A_j)\) be a finite family of normalized periodic tensors with arbitrary periods, and let \(\mu _j\ne 0\). Assume explicitly that their original middle ground spaces \((\mathcal G_{A_j}(n))_j\) are independent at every sufficiently large length \(n\). For \(T=\bigoplus _j\mu _jA_j\), there exists \(R_0{\gt}0\) such that
In particular, the canonical local projection of range \(R_0\) gives a positive interaction whose open kernel is exactly \(\mathcal G_T(N)\) for every \(N\ge R_0\). The family may be empty. This is an independence criterion: its middle-space hypothesis remains explicit and is not an unrestricted periodic-family existence theorem.
Each periodic sector has its original one-step intersection identity eventually. Finiteness and the assumed middle independence allow the geometric joint-intersection criterion to combine these identities. Nonzero weights preserve the sum of the sector spaces. Iterating the joint intersections gives the exact canonical kernels.
Under the preceding hypotheses, one can choose a positive interaction \(h\) of range \(R{\gt}0\), with exact open kernels for all \(N\ge R\), and one constant \(\gamma {\gt}0\) as follows. For every pure normalized zero-energy state \(\varphi \) of \(h\) and every centered local observable \(X\) on a positive-length interval beginning at \(a\), the literal finite-volume commutator energies converge along every interval exhaustion whose left and right margins diverge:
The constant is chosen before the state, observable, filter, and boundary margins. No translation-invariance assumption is imposed on the state being classified. The middle-space independence assumption is retained in this sufficient criterion.
Apply the existing original-chain periodic-family gap theorem to the constructed interaction. Its eventual kernel hypothesis follows from the exact identities for all volumes at least the interaction range.
Let \(A\) be any tensor and \(R,L{\gt}0\). Assume that its canonical open Hamiltonian has kernel \(\mathcal G_A(N)\) for every \(N\geq R\) and one positive norm gap at every sufficiently large length divisible by \(L\). Assume also that, for every choice of exterior lengths \(K(M),r(M)\), the projection defect of the cover with left length \((K(M)+M)L\), right length \(ML+r(M)\), and overlap length \(ML\) tends to zero as \(M\to \infty \). Then there is \(\delta {\gt}0\) such that
All three estimates are supplied hypotheses of this criterion. This is the covering argument of [ 225 , Section 6 ] .
Convergence along every selection gives one overlap length at which every exterior choice has projection defect at most \(1/2\). Increase that overlap length to lie beyond the aligned-gap threshold and the interaction range. The two-interval comparison gives a common gap at every sufficiently large original length. A positive finite minimum includes the remaining lengths, with their actual kernels.
If two tensors have the same \(R\)-site boundary space, their canonical interaction matrices agree. For \(R{\gt}0\), their canonical open Hamiltonians consequently agree at every volume. Their bond dimensions may differ.
The canonical matrix represents the orthogonal projection onto the local boundary-space complement. Equality of that space therefore gives equality of the matrices and of every sum of their local translates, as in [ 225 , Lemma existenceinteraction ] .
Suppose that \(V^*V=I\) and \(V^*A_i=B_iV^*\) for every physical letter. Then \(\mathcal G_N(B)\subseteq \mathcal G_N(A)\) for every \(N\geq 0\). No normalization or injectivity assumption is required.
The letter relation extends to words. Trace cyclicity gives \(\Gamma ^A_N(VYV^*)=\Gamma ^B_N(Y)\), proving the range inclusion.
Let \(A,A'\) be gauge and phase inequivalent normalized irreducible periodic tensors, with possibly different periods and bond dimensions. Fix \(L{\gt}0\) and tensors \(B,C\) on \(L\)-site blocks, together with isometries \(V,W\) satisfying \(V^*(A^{[L]})_i=B_iV^*\) and \(W^*(A'^{[L]})_i=C_iW^*\) for every blocked letter. Suppose that \(B\) is a primitive normalized tensor of positive bond dimension with a faithful invariant matrix. Then \(B,C\) are inequivalent up to similarity and phase whenever their bond dimensions agree. The conclusion follows from original-chain separation; separation of compressed corners is not assumed.
A hypothetical corner gauge relation gives \(\mathcal G_N(B)=\mathcal G_N(C)\) at every \(N\). For large \(N\), primitivity and faithfulness make \(\Gamma ^B_N\) injective, so \(x=\Gamma ^B_N(I)\) is nonzero. The two isometries place \(x\) in both blocked original spaces. The common physical blocking isometry gives a nonzero \(y\in \mathcal G_{NL}(A)\cap \mathcal G_{NL}(A')\). Original-chain overlap decay at coefficient \(1/2\) implies \(\| y\| ^2\leq \tfrac 12\| y\| ^2\), a contradiction.
Let \(A\) satisfy \(\sum _iA_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace with \(E_A(\rho )=\rho \). There exist a positive dimension \(E\), an isometry \(V\), a normalized tensor \(B\), and a faithful stationary matrix \(\sigma {\gt}0\) such that
Let \(\varphi \) be the quasi-local state defined by the original generators \(A,\rho \). For every \(k\geq 0\), there is a matrix \(\tau _k\geq 0\) of trace one such that, for every translated interval \([a,a+k)\) and every matrix observable \(X\) on that interval,
The physical dimension is derived from the compressed normalization. No faithfulness of \(\rho \) or primitivity of the generating tensor is required. This concerns supplied stationary generating data in [ 225 , Section 3 ] and its finite support identification in Section 4; constructing such data from an arbitrary GVBS boundary limit remains separate [ 61 ] .
Support compression gives \(E{\gt}0\), faithful stationary generators \(B,\sigma \), and \(\eta _{A,\rho }(X)=\eta _{B,\sigma }(X)\) at every length. Normalization of \(B\) implies positive physical dimension. The faithful state has a density \(\tau _k\) of range \(G_B(k)\) on every translated interval. The interval evaluation formula for the original quasi-local state identifies its restriction with \(\eta _{A,\rho }\), proving (547).
Let \(d\geq 1\), let \(\sum _iA_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace and satisfy \(E_A(\rho )=\rho \). For every range \(R\geq 0\), the quasi-local state defined by \(A,\rho \) belongs to the zero-energy face of the canonical interaction determined by \(G_A(R)\). No faithfulness or purity is assumed.
The canonical interaction annihilates \(G_A(R)\). The finite insertion expectation of every translate of that interaction is consequently zero, so the generated state is a zero-energy state, as in [ 225 , Lemma existenceinteraction ] .
Let \(d\ge 1\), let \((A_j)\) be a finite family of normalized periodic tensors with arbitrary periods, and let \(\mu _j\ne 0\). For \(T=\bigoplus _j\mu _jA_j\), there exists \(R_0{\gt}0\) such that
In particular, the canonical local projection of range \(R_0\) defines a positive interaction with exact open kernels at every \(N\ge R_0\). No inequivalence or independence of the supplied sectors is assumed. The finite family may be empty. This is the finite periodic-tensor presentation of the interaction construction in Section 4 of [ 225 ] ; the passage from general GVBS data to such a presentation remains separate.
Select inequivalent periodic representatives without changing any finite joint ground space. Their middle ground spaces are eventually independent, so the joint-intersection criterion constructs exact canonical kernels. Equality of the local support spaces transports the canonical interaction back to the original weighted family.
Under the preceding hypotheses, there are a positive interaction \(h\) of range \(R{\gt}0\), with exact open kernels for all \(N\ge R\), and a constant \(\gamma {\gt}0\) such that every pure normalized zero-energy state \(\varphi \) and every centered local observable \(X\) on a positive-length interval beginning at \(a\) satisfy
The limit holds along every interval exhaustion with both margins diverging. The constant precedes the state, observable, filter, and margins. Neither translation invariance nor supplied interaction, kernel, separation, or independence data are required. The statement concerns the finite periodic-tensor presentation; the general GVBS presentation step remains separate.
Apply the periodic-family original-chain commutator theorem to the constructed interaction. Its eventual kernel identity follows from the exact kernels for every volume at least the interaction range.
Let \(A\) have positive bond dimension, satisfy \(\sum _iA_i^*A_i=I\), and have a faithful stationary matrix \(\rho {\gt}0\). There exists \(R_0{\gt}0\) such that its canonical open Hamiltonians satisfy
In particular, the canonical local projection of range \(R_0\) defines a positive parent interaction with these exact kernels. The tensor need not be irreducible or periodic, and no interaction or kernel identity is supplied. This concerns the normalized generating data in [ 225 , Section 3 ] ; their construction from an arbitrary GVBS boundary limit remains separate.
The faithful stationary matrix makes every invariant projection reducing. Decompose into normalized irreducible periodic corners, preserving every boundary space. Construct the periodic-family canonical interaction. Equality of the local supports then transfers the canonical matrices and open Hamiltonians back to \(A\).
Under the preceding hypotheses, there exist \(R{\gt}0\), a positive interaction \(h\), and \(\gamma {\gt}0\) with exact open kernels \(G_A(N)\) for all \(N\geq R\). The state defined by \(A,\rho \) has zero energy for \(h\). Every pure normalized zero-energy state \(\varphi \) and centered local observable \(X\) on a positive-length interval satisfy
The literal limit holds along every interval exhaustion with both margins diverging. The interaction and constant precede the state, observable, filter, and margins. No translation invariance is imposed on the competing pure states.
Choose the canonical interaction given by the preceding theorem. Its local projection annihilates the generated state support, so that state has zero energy. Its exact kernels supply the eventual kernel identity in the faithful-generator commutator theorem.
Let \(A\) be a normalized primitive tensor of positive bond dimension with stationary matrix \(\rho \). If \(\rho \) is positive definite, then blocking by any positive length retains primitivity with the same stationary matrix \(\rho \). A bijective relabelling of the physical letters also preserves the primitive witness with the same \(\rho \); this second assertion requires only the original primitive fixed-point data.
Positive blocking preserves trace preservation, irreducibility, and peripheral primitivity. The same \(\rho \) is stationary for every positive power of the transfer map. The complementary fixed-point spectral gap therefore holds at this supplied stationary matrix. Under a bijective physical relabelling, both the normalization sum and the transfer map are unchanged.
Let \(d{\gt}0\), let \(A,B\) be trace-preserving tensors with possibly different bond dimensions, and let \(\rho ,\sigma \geq 0\) be stationary matrices of nonzero trace. If \(\eta _{A,\rho }(X)=\eta _{B,\sigma }(X)\) for every observable at every finite length, then their continuous quasi-local expectation functionals satisfy \(\omega _{A,\rho }=\omega _{B,\sigma }\). Faithfulness and purity are unnecessary. This concerns the generating expectations in [ 225 , Section 3 ] .
Embed any finite region into its canonical containing interval. Its expectation is the insertion value of that interval observable, so the two finite-region functionals agree. Uniqueness of continuous extension from the finite-region algebras gives equality on the quasi-local algebra.
Let \(A\) satisfy \(\sum _iA_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace with \(E_A(\rho )=\rho \). There exist a positive dimension \(E\), an isometry \(V\), a normalized tensor \(B\), and a faithful stationary matrix \(\sigma {\gt}0\) with
The physical dimension is positive, and the actual continuous functionals agree: \(\omega _{A,\rho }=\omega _{B,\sigma }\). No purity or minimality is assumed. This is a consequence for supplied stationary generating data in [ 225 , Section 3 ] . Their construction from arbitrary GVBS boundary limits remains separate [ 61 ] .
Compress the generators and stationary matrix to their stationary support. The compressed bond dimension is positive, and normalization of \(B\) implies positive physical dimension. All finite insertion expectations agree under compression. The preceding equality theorem therefore identifies their continuous extensions.
Let \(A\) have positive bond dimension and let its blocking by \(L\) have a finite isometric resolution into primitive tensors \(B_j\) of positive bond dimensions with faithful stationary matrices \(\rho _j\) and pairwise inequivalent sectors. Suppose the embeddings satisfy
Assume that for every \(j,r\) the matrix
is an orthogonal projection. For \(N=KL+ML+r\), let \(P_N\) project onto \(\mathcal G_A(N)\), let \(P_{\mathrm L}\) project onto \(\mathcal G_A(KL+ML)\otimes \mathcal H_r\), and let \(P_{\mathrm R}\) project onto \(\mathcal H_{KL}\otimes \mathcal G_A(ML+r)\). Along every filter for which \(M\to \infty \), with \(K,r\) arbitrary,
No periodicity assumption on \(A\) is imposed. All sector-resolution and tail-Gram hypotheses are supplied explicitly.
The three joint sector spaces equal the corresponding original interval spaces under one configuration isometry. Their joint projection defect converges to zero by primitive-sector decay and supported cancellation. Isometric conjugation preserves its norm; taking the adjoint reverses the two interval projections.
Assume the preceding hypotheses and \(R,L{\gt}0\). Suppose that the canonical range-\(R\) open Hamiltonian has kernel \(\mathcal G_A(N)\) for every \(N\ge R\), and has a common positive norm gap at all sufficiently large lengths divisible by \(L\). Then there is \(\delta {\gt}0\) such that, for every \(N\) and every \(v\perp \ker H_{R,N}^{\mathrm{open}}(A)\),
Exact kernels and the aligned gap remain explicit assumptions of this criterion. The conclusion uses the actual kernel also at volumes shorter than the interaction range.
The first theorem supplies the residual-cover limit uniformly along all prefix and tail selections. The aligned-cover criterion gives an eventual original-length gap. A positive minimum over the remaining finite set of volumes gives one bound at every length.
Let \(J_j\) be the coordinate inclusion of the \(j\)th summand of a finite direct sum, and let \(T_i=\bigoplus _j\mu _j A_j^i\). Then
The statements permit summands of dimension zero and an empty family. They are the coordinate identities underlying the canonical direct-sum decomposition in [ 13 ] , Section II.
The unflattened coordinate inclusions resolve the identity of the dependent direct sum and intertwine every block diagonal matrix with its selected block. Transport these identities through the common coordinate equivalence.
Let \(A\) satisfy \(\sum _i A_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace and satisfy \(\sum _i A_i\rho A_i^*=\rho \). There exist a tensor \(B\) of positive bond dimension and a faithful stationary matrix \(\sigma {\gt}0\) such that
In particular, the physical dimension is positive. There also exist a range \(R{\gt}0\), a positive interaction \(h\), and \(\gamma {\gt}0\) such that
Here \(\mathcal F_h\) denotes the normalized positive states with zero expectation of every translate of \(h\). For every pure \(\varphi \in \mathcal F_h\), every local observable \(X\) on a positive-length interval \([a,a+k)\) with \(\varphi (X)=0\), and every interval exhaustion whose left and right margins both diverge,
The interaction and constant are chosen before the pure state, observable, filter, and margins. The generated state may be mixed; neither its purity nor translation invariance of the competing states is required. The finite kernels are \(G_B(N)\); equality with \(G_A(N)\) is not asserted.
This is a consequence for supplied stationary generating data in [ 225 , Section 3 ] . Their construction from arbitrary GVBS boundary limits remains separate, as documented in [ 61 ] .
Compress the stationary matrix to its support. This gives normalized faithful stationary generators for the same quasi-local state. Apply the constructed parent-interaction theorem to those generators. Equality of the state functionals transfers zero-energy membership to the original generated state. The interaction, its exact kernels, and its common commutator gap are retained.
Suppose \(\sum _i A_i^*A_i=\mathbf1\) and \(P_{j+1}A_i=A_iP_j\), with labels in \(\mathbb Z/m\mathbb Z\) for some \(m{\gt}0\). Then, for every \(r\geq 0\),
Replace the label \(j\) by \(-j\) and apply the forward cyclic word identity.
Let \(A\) be a normalized irreducible periodic tensor of period \(m\), and let \(L{\gt}0\) be divisible by \(m\). There are positive-dimensional primitive tensors \(B_j\) on the \(L\)-site alphabet, faithful stationary matrices \(\rho _j\), cyclic projections \(P_j\), and isometries \(V_j\) such that
The tensors \(B_j\) are pairwise inequivalent up to similarity and phase. For every tail length \(r\), their literal support projections satisfy \(\sum _{|\tau |=r}A_\tau ^*V_jV_j^*A_\tau =P_{j-r}\).
Write \(L=mq\). Power each minimum-period primitive sector by \(q\) and identify the iterated physical alphabet with the \(L\)-site alphabet. This preserves the stationary matrices and both intertwiners. Separation of the powered sectors follows by applying original-chain separation to the distinct period-one sectors. The last identity follows from the cyclic word Gram.
Let \(h\) be a matrix on an \(R\)-site interval. Its matrix sum over all nonwrapping windows and the corresponding linear operator both vanish when the chain length satisfies \(N{\lt}R\). If \(R{\gt}0\), the matrix and linear operator representations of the open interaction sum agree for every \(N\geq 0\). No positivity assumption on \(h\) is needed. This is the empty-sum convention in [ 225 ] , equation (3.12).
For \(N{\lt}R\) there is no nonwrapping window of length \(R\), so both sums are empty. For \(N\geq R\), the coordinate representation intertwines each local term, and linearity identifies their sums.
Let \((A_j)\) be a finite family of pairwise inequivalent normalized irreducible periodic tensors, with periods \(m_j\), and let \(L{\gt}0\) be divisible by every \(m_j\). Put \(\Phi _i=\bigoplus _j(A_j)_i\). There is a family of positive-dimensional primitive tensors \((B_s)\) on the \(L\)-site alphabet, indexed by pairs \(s=(j,a)\) with \(a\in \mathbb Z/m_j\mathbb Z\), with faithful stationary matrices \(\rho _s\) and isometries \(V_s\) into the original direct sum, such that
The tensors \(B_s\) are pairwise inequivalent up to similarity and phase. For every \(s\) and every tail length \(r\geq 0\), the matrix
is an orthogonal projection. The family may be empty.
Choose the literal powered cyclic resolution of each original tensor and compose its isometries with the coordinate inclusions of the direct sum. The inclusions resolve the identity and preserve both intertwiners. Within one original tensor, separation is part of the powered resolution; between different original tensors, it follows from original-chain angle decay. Finally, if \(s=(j,a)\), the word identity gives \(Q_s(r)=J_jP_{j,a-r}J_j^*\). This is an orthogonal projection because \(J_j\) is an isometry and \(P_{j,a-r}\) is a cyclic projection.
Let \(A\) be a tensor and let \(R{\gt}0\). Suppose the canonical open Hamiltonian has kernel \(\mathcal G_A(N)\) for every \(N\ge R\). Then the range-\(R\) canonical orthogonal-complement projection is a positive interaction \(h\), and the open interaction generated by \(h\) has those same exact kernels. The interaction is literally the configuration-basis matrix of that canonical projection.
The canonical orthogonal-complement projection is positive. Its configuration-basis matrix represents the same local operator, hence gives the same open Hamiltonian and kernel at every volume.
Let \(A\) be a tensor. Suppose that, for every sufficiently large range \(W\), its canonical open Hamiltonians have kernels \(\mathcal G_A(N)\) for \(N\ge W\) and a positive uniform norm gap above their actual kernels at all volumes. Let \(h\ge 0\) have positive range \(R\), and suppose \(\ker H_h(N)=\mathcal G_A(N)\) for all sufficiently large \(N\). Then there exists \(\gamma {\gt}0\) such that, for every \(N\ge 0\) and \(v\perp \ker H_h(N)\),
No local kernel condition or tensor injectivity is imposed.
Choose \(W\ge R\) large enough for both kernel identities and the canonical gap. Positivity and finite dimensionality give \(H_h(W)\ge \kappa P_{\mathcal G_A(W)^\perp }\) with \(\kappa {\gt}0\). Sum this comparison over all \(W\)-site windows. Their multiplicity is at most \(W-R+1\), so the canonical gap \(\delta _W\) gives the positive bound \(\kappa \delta _W/(W-R+1)\) on all \(N\ge W\). Take its minimum with the finitely many positive gaps at \(N{\lt}W\), each measured above the actual kernel.
Let \(V:\mathbb C^D\to \mathbb C^d\otimes \mathbb C^D\) be a linear map, with the physical coordinate written first. Define its physical matrix slices by
This is the coordinate convention for the supplied representation in [ 225 , equation (3.6) ] .
For the preceding slices,
Thus an isometry gives trace-preserving normalized tensor matrices. The insertion identity holds for any supplied \(V\), without an isometry assumption. No dimension, stationarity, or faithfulness assumption is needed. This identifies the one-site coordinates of [ 225 , equation (3.6) ] .
Expand the matrix products in physical and virtual coordinates. The normalization is the Gram matrix of \(V\). Permuting the finite sums in the insertion gives the displayed two-sided Kraus expression.
Identify configurations on one site with their physical letter. For any tensor \(A\), its one-site physical observable transfer is
For the slices of \(V\), this is trace-dual to the preceding Heisenberg insertion:
The matrices \(X,B,\rho \) are arbitrary. These identities neither construct a supplied isometry from arbitrary GVBS data nor identify multi-site expectations with a particular ordering of source sites.
Reindex the one-site configuration sum by its unique physical letter. Cyclicity of the trace turns each term \(X_{ij}\operatorname {Tr}(\rho A_i^*BA_j)\) into \(X_{ij}\operatorname {Tr}(BA_j\rho A_i^*)\).
Let \(A\) have positive bond dimension and satisfy \(\sum _i A_i^*A_i=I\). Suppose that \(\rho {\gt}0\) and \(\sum _i A_i\rho A_i^*=\rho \). There is \(R_0{\gt}0\) such that, for each \(R\geq R_0\), the canonical open Hamiltonians satisfy
and there is \(\delta _R{\gt}0\) such that
The constant is independent of the original chain length, including its residue modulo every blocking length. This is the supplied generating-data case of [ 225 ] , Theorems 1.1–1.2.
Decompose the faithful stationary tensor into a finite periodic family, with equality of the boundary spaces at every length. The uniform family gap gives a common range threshold. Equality of the local support spaces identifies the canonical interactions and all their open Hamiltonians. Transport the exact kernels and the norm gap through these identities.
Under the preceding hypotheses, there are \(R{\gt}0\), a positive interaction \(h\) of range \(R\), and constants \(\delta ,\gamma {\gt}0\) such that
The generated stationary state belongs to the zero-energy face of \(h\). Every pure state \(\varphi \) in this face and every observable \(X\) on an interval of positive length with \(\varphi (X)=0\) satisfy
The limit holds along every interval exhaustion for which both margins around the support of \(X\) diverge. The choices of \(R,h,\delta ,\gamma \) precede the state, observable, and exhaustion. No purity of the generated stationary state is assumed. This is the supplied generating-data case of [ 225 ] , Theorems 1.1–1.2 and Section 6.
Choose the canonical interaction at the uniform threshold. The first theorem gives its exact kernels and finite-volume gap. The generated state has zero energy for this canonical interaction. Its eventual exact kernel identity also gives the common literal commutator gap in every pure zero-energy state.
Let \(d\ge 1\), let \((A_j)\) be a finite family of normalized periodic tensors with arbitrary periods, and let \(\mu _j\ne 0\). For \(T=\bigoplus _j\mu _jA_j\), there exists \(R_0{\gt}0\) such that every \(R\ge R_0\) has both of the following properties:
and there exists \(\delta _R{\gt}0\) such that, for every \(N\ge 0\) and every \(v\perp \ker H_{R,N}^{\mathrm{open}}(T)\),
The finite family may be empty. No inequivalence, invariant matrix, primitive presentation, interaction, or kernel identity is supplied. The norm bound uses the actual kernel also when \(N{\lt}R\). This is the finite periodic-tensor presentation of the gap theorem; the passage from general GVBS data to such a presentation remains separate.
Select inequivalent representatives without changing any finite ground space. If the selected family is nonempty, a common period multiple gives a literal primitive isometric resolution and exact orthogonal tail Gram projections. The aligned finite gap and the residual-cover limit extend the gap to every original length. For an empty selected family, the aligned theorem applies with blocking length one. Equality of local supports transports the canonical gap to the original weighted family.
Under the preceding hypotheses, one can choose \(R{\gt}0\) and the canonical positive interaction \(h\) of range \(R\) so that its open kernels are \(\mathcal G_T(N)\) for every \(N\ge R\). There exists \(\delta {\gt}0\) giving a norm gap above the actual open kernel at every original volume, including \(N{\lt}R\). For the same interaction there exists \(\gamma {\gt}0\) such that every pure normalized zero-energy state \(\varphi \) and every centered local observable \(X\) on a positive-length interval satisfy
The limit holds along every exhaustion with both margins diverging. Both constants precede the states, observables, filters, and margins. No translation invariance of a classified state is assumed. The general GVBS presentation step remains separate.
Choose any range at the canonical threshold and use its canonical local projection matrix. The all-length canonical norm gap transfers to the matrix interaction sum, including short volumes. The exact kernels supply the eventual-kernel hypothesis of the original-chain periodic-family commutator theorem.
Let \(A\) satisfy \(\sum _i A_i^*A_i=I\), and let \(\rho \geq 0\) have nonzero trace and satisfy \(\sum _i A_i\rho A_i^*=\rho \). There exist a tensor \(B\) of positive bond dimension and a faithful stationary matrix \(\sigma {\gt}0\) such that
In particular, the physical dimension is positive. There also exist a range \(R{\gt}0\), a positive interaction \(h\), and constants \(\delta ,\gamma {\gt}0\) such that
The same interaction satisfies the finite-volume estimate
Here \(\mathcal F_h\) denotes the normalized positive states with zero expectation of every translate of \(h\). For every pure \(\varphi \in \mathcal F_h\), every local observable \(X\) on a positive-length interval \([a,a+k)\) with \(\varphi (X)=0\), and every interval exhaustion whose left and right margins both diverge,
The interaction and constant are chosen before the pure state, observable, filter, and margins. The generated state may be mixed; neither its purity nor translation invariance of the competing states is required. The finite kernels are \(G_B(N)\); equality with \(G_A(N)\) is not asserted.
This is a consequence for supplied stationary generating data in [ 225 , Section 3 ] . Their construction from arbitrary GVBS boundary limits remains separate, as documented in [ 61 ] .
Compress the stationary matrix to its support. This gives normalized faithful stationary generators for the same quasi-local state. Apply the constructed parent-interaction theorem to those generators. Equality of the state functionals transfers zero-energy membership to the original generated state. The interaction, its exact kernels, its uniform finite-volume gap, and its common commutator gap are retained.
Let \(D{\gt}0\) and let \(V:\mathbb C^D\to \mathbb C^d\otimes \mathbb C^D\) be an isometry, with tensor slices \(A_i\). Then \(\sum _iA_i^*A_i=I\), and the transfer map of \(A\) has a positive fixed point \(\rho \) of trace one. There are a faithful normalized tensor \(B\) with a positive definite fixed point that generates the same quasi-local state, a range \(R{\gt}0\), and a positive interaction \(h\) on \(R\) sites whose open-chain kernels equal \(G_N(B)\) for \(N\geq R\), such that the generated state has zero energy, the open Hamiltonians of \(h\) have one positive gap at every length, and one constant \(\gamma {\gt}0\) bounds the commutator energy below in every pure zero-energy state, for every centered observable on an interval of positive length and every interval exhaustion whose left and right margins both diverge [ 225 , equation (3.6), Theorems 1.1–1.2, and Section 6 ] . The isometry is supplied; its construction from a general presentation is not asserted [ 61 ] .
The isometry condition is the trace-preserving normalization, so the transfer channel has a fixed density matrix. Compressing to its support gives faithful generators of the same state, and the stationary faithful-generator theorem supplies the interaction and both gaps.
Let \(A\) be normalized with positive bond dimension and a faithful stationary matrix, as in Theorem 12.9.315. Let \(h\geq 0\) be any given interaction of positive range \(R\) such that \(\ker H_h(N)=G_A(N)\) for all sufficiently large \(N\). Then there is \(\delta {\gt}0\) such that
The local kernel \(\ker h\) need not equal \(G_A(R)\). This is the finite-volume conclusion of [ 225 ] , Theorem 1.2, in the supplied faithful generating-data setting.
Choose a sufficiently long window \(W\) with exact kernel \(G_A(W)\) and a uniform canonical gap at range \(W\). Positivity gives a local lower bound \(H_h(W)\geq \kappa P_{G_A(W)^\perp }\) with \(\kappa {\gt}0\). Summing these windows counts each original term at most \(W-R+1\) times. This transfers the canonical gap to all sufficiently long chains. The finitely many remaining volumes are included by a positive finite minimum above their actual kernels.
Under the preceding hypotheses, the same given interaction \(h\) has a uniform finite-volume gap at every original chain length and one common literal infinite-volume commutator gap in every pure zero-energy state. The infinite-volume limit holds along every interval exhaustion with both margins around the local observable diverging. Both positive constants precede the state, observable, and exhaustion. This is the given-interaction conclusion of [ 225 ] , Theorem 1.2, in the supplied faithful generating-data setting.
Apply the preceding finite-volume theorem and the faithful-generator commutator theorem to the same interaction and its eventual kernel identity.
For a finite list of one-site observables \((X_1,\ldots ,X_n)\), define \(O_{\mathrm{rev}}=X_n\otimes \cdots \otimes X_1\). The empty product is the identity on the zero-site space. Equivalently, \(O_{\mathrm{rev}}(X_1,X_2,\ldots ,X_n) =O_{\mathrm{rev}}(X_2,\ldots ,X_n)\otimes X_1\). This records the physical order required by trace duality of the source-written Heisenberg composition in [ 225 , equations (3.5)–(3.6) ] .
Let \(V:\mathbb C^D\to \mathbb C^d\otimes \mathbb C^D\) be a supplied map, let \((A_i)_{ab}=V_{(i,a),b}\), and set \(F_X(B)=V^*(X\otimes B)V\). For every matrix \(\rho \) and every finite list \((X_1,\ldots ,X_n)\),
Here \(E\) is the physical observable transfer with forward matrix words. The equality includes \(n=0\). It holds in particular for an isometry, whose slices satisfy \(\sum _iA_i^*A_i=I\); neither normalization, positivity, nor stationarity is needed for the trace identity itself. This is the explicit order relation for the supplied representation in [ 225 , equations (3.5)–(3.6) ] , and makes no assertion that arbitrary GVBS boundary data have this representation.
Induct on the list, allowing an arbitrary matrix \(\rho \). Trace duality moves the first Heisenberg insertion to \(E_{X_1}(\rho )\). The induction hypothesis then gives \(\operatorname {Tr}(E_{O_{\mathrm{rev}}(X_2,\ldots ,X_n)} (E_{X_1}(\rho )))\). The inserted-transfer product identity identifies this with the expectation of \(O_{\mathrm{rev}}(X_2,\ldots ,X_n)\otimes X_1\). At the empty list, both expressions are \(\operatorname {Tr}\rho \).
Let \(d\ge 1\), let \((A_j)\) be a finite family of normalized periodic tensors, let \(\mu _j\ne 0\), and write \(T=\bigoplus _j\mu _jA_j\). Let \(h\ge 0\) be a supplied interaction of range \(R{\gt}0\) whose open Hamiltonians satisfy \(\ker H_N(h)=\mathcal G_T(N)\) for every sufficiently large \(N\). Then there exists \(\delta {\gt}0\) such that every original volume \(N\) and every \(v\perp \ker H_N(h)\) satisfy
The actual kernel is used also on short intervals. No local-kernel identity at range \(R\), lower bound on \(R\) beyond positivity, sector separation, stationary matrix, or finite gap is supplied. The finite family may be empty. The statement concerns finite periodic-tensor presentations; the general GVBS presentation step remains separate. This is the given-interaction conclusion of [ 225 , Theorem 1.2 ] in this setting.
The canonical interactions have exact kernels and uniform gaps at every sufficiently large range. Choose one long interval on which both the canonical and supplied interaction kernels equal the MPS space. Finite-dimensional positivity bounds the excitation projection below the supplied Hamiltonian. The window comparison transfers the canonical gap, and a positive minimum includes the remaining short volumes above their actual kernels.
Under the preceding hypotheses, the same supplied interaction \(h\) has an all-length finite norm gap with a constant \(\delta {\gt}0\) and a constant \(\gamma {\gt}0\) such that every pure normalized zero-energy state \(\varphi \) and every centered local observable \(X\) on a positive-length interval beginning at \(a\) satisfy
The limit holds along every exhaustion with both margins diverging. Both constants precede the states, observables, filters, and margins. No translation invariance of a classified state is assumed. The general GVBS presentation step remains separate.
Combine the given-interaction finite gap with the original-chain periodic-family commutator theorem, applied to exactly the same supplied interaction and eventual kernel identity.
12.10 Continuity of the parent-Hamiltonian gap
Along a continuous path of tensors, the phase argument of [ 236 , Appendix A, lines 2475–2580 ] needs a gap of the parent Hamiltonians that is uniform both in the path parameter and in the chain length. The source obtains it from Nachtergaele’s bound, whose finite-size input is the gap of the Hamiltonian restricted to a fixed number of sites; that gap has a uniform lower bound because the restricted Hamiltonian depends continuously on the parameter [ 236 , lines 2575–2578 ] . This section proves the continuity step and combines it with Knabe’s nearest-neighbor criterion and compactness. The single-block results apply to normal tensors with a common injectivity length \(p\), at every fixed interaction range \(R{\gt}p\). Quantum Wielandt supplies \(p=D^4\) from normality alone. The sharper one-site injective result has range two. The source’s path also covers normal forms with several blocks; the scope of the compact-gap conclusions is recorded in [ 176 ] .
Let \(H,H_0,P,P_0\) be bounded operators on a complex normed space, and suppose \(\delta \lVert v-P_0v\rVert \leq \lVert H_0v\rVert \) for all \(v\), with \(\delta \geq 0\), \(\lVert H-H_0\rVert \leq \varepsilon \) and \(\lVert P-P_0\rVert \leq r\). Then \((\delta (1-r)-\varepsilon )\lVert v\rVert \leq \lVert Hv\rVert \) for every \(v\) with \(Pv=0\). Consequently, if \(x\mapsto H_x\) and the orthogonal projection onto a subspace \(K_x\) are continuous at \(x_0\), \(K_{x_0}\subseteq \ker H_{x_0}\), and \(\delta \lVert v\rVert \leq \lVert H_{x_0}v\rVert \) on \(K_{x_0}^\perp \) with \(\delta {\gt}0\), then every \(\delta '{\lt}\delta \) satisfies \(\delta '\lVert v\rVert \leq \lVert H_xv\rVert \) on \(K_x^\perp \) for all \(x\) near \(x_0\).
For \(Pv=0\) one has \(\lVert P_0v\rVert =\lVert (P-P_0)v\rVert \leq r\lVert v\rVert \) and \(\lVert H_0v\rVert \leq \lVert Hv\rVert +\varepsilon \lVert v\rVert \); combining these with the hypothesis gives the first claim. For the second, the gap on \(K_{x_0}^\perp \) and \(K_{x_0}\subseteq \ker H_{x_0}\) give \(\delta \lVert v-P_{x_0}v\rVert \leq \lVert H_{x_0}v\rVert \) for all \(v\), and one chooses \(\varepsilon =(\delta -\delta ')/2\) and \(r=\varepsilon /\delta \).
Let \(A\) be injective at a length \(p{\gt}0\), and let \(R{\gt}p\). There is \(\delta {\gt}0\) such that
In particular, for a normal tensor of bond dimension \(D{\gt}0\), every fixed \(R\geq D^4+1\) has this property. No normalization is required.
Injectivity propagates to length \(R-1\). The long-chain martingale estimate and range comparison give a positive gap at range \(R\). The exact open kernel identity and finite dimensionality handle the finitely many shorter admissible chains. Quantum Wielandt supplies \(p=D^4\) for a normal tensor.
Let \(E\) be a finite-dimensional Hilbert space, \(F\) a Hilbert space, and \(T_x\colon E\to F\) a continuous family of injective bounded maps. Then the orthogonal projection \(T_x(T_x^\dagger T_x)^{-1}T_x^\dagger \) onto the range of \(T_x\) depends continuously on \(x\).
The Gram operator \(T_x^\dagger T_x\) is invertible and continuous in \(x\), and inversion is continuous at invertible elements of a Banach algebra.
Let \(x\mapsto A_x\) be a continuous family of tensors. For fixed \(L\), the boundary map \(X\mapsto \sum _\sigma \operatorname {tr}(A_{x,\sigma }X)\lvert \sigma \rangle \) on \(L\) sites is continuous in \(x\). If every \(A_x\) is \(L\)-block injective, the orthogonal projection onto the local ground space on \(L\) sites is continuous in \(x\). Hence, for \(L\leq N\), the open and periodic parent Hamiltonians with interaction range \(L\) on \(N\) sites are continuous in \(x\).
Each matrix entry of the boundary map is a trace of a fixed product of tensor letters. Block injectivity makes the boundary map injective, and its range is the local ground space, so the previous theorem applies. The Hamiltonians are finite sums of translates of one minus that projection.
Let \(x\mapsto A_x\) be continuous, \(R\leq N\), and suppose every \(A_x\) is \(R\)-block and \(N\)-block injective. If \(\delta \lVert v\rVert \leq \lVert H^{\mathrm{open}}_{N,R}(A_{x_0})v\rVert \) for all \(v\perp G_N^{\mathrm{ES}}(A_{x_0})\), with \(\delta {\gt}0\), then for every \(\delta '{\lt}\delta \) and every \(x\) near \(x_0\), \(\delta '\lVert v\rVert \leq \lVert H^{\mathrm{open}}_{N,R}(A_x)v\rVert \) for all \(v\perp G_N^{\mathrm{ES}}(A_x)\). This is the continuity step of [ 236 , lines 2575–2578 ] .
The open parent Hamiltonian annihilates the ground space, and both it and the ground-space projection are continuous in \(x\).
Let \(0{\lt}R\leq W\leq N\). If \(\kappa \, h_W\leq H^{\mathrm{open}}_{W,R}\) for the range-\(W\) parent interaction \(h_W\) and some \(\kappa {\gt}0\), then \(\kappa H^{\mathrm{open}}_{N,W}\leq (W-R+1)H^{\mathrm{open}}_{N,R}\). For a tensor for which both open kernels equal \(G_N^{\mathrm{ES}}\), a gap \(\gamma \lVert v\rVert \leq \lVert H^{\mathrm{open}}_{N,W}v\rVert \) on \((G_N^{\mathrm{ES}})^\perp \) gives \(\frac{\kappa \gamma }{W-R+1}\lVert v\rVert \leq \lVert H^{\mathrm{open}}_{N,R}v\rVert \) on the same complement. The assertion may be used at one volume or uniformly over every \(N\geq W\). For a one-site injective tensor with \(R=2\) and \(W\geq 2\), the kernel hypotheses hold automatically.
Translate the local comparison to every window of \(W\) consecutive sites and sum; each range-\(R\) term lies in at most \(W-R+1\) windows. Both open Hamiltonians have kernel \(G_N^{\mathrm{ES}}\), so the operator inequality transfers the gap.
Let \(A\) be a one-site injective tensor of nonzero bond dimension. Then the open-chain Hamiltonian \(H^{\mathrm{open}}_{N,2}(A)\) built from the canonical two-site parent interaction has a positive gap above \(G_N^{\mathrm{ES}}(A)\), uniform in all sufficiently large \(N\). In particular, there are \(m\geq 2\) and \(\gamma \) with \(m\gamma {\gt}1\) such that \(\gamma \lVert v\rVert \leq \lVert H^{\mathrm{open}}_{m+1,2}(A)v\rVert \) for all \(v\perp G_{m+1}^{\mathrm{ES}}(A)\).
Gauge \(A\) to a primitive tensor with a positive-definite fixed point, which has the same local ground spaces, and take its open-chain gap \(\gamma _W\lVert v\rVert \leq \lVert H^{\mathrm{open}}_{N,W}(A)v\rVert \) at some range \(W\geq 2\) for all \(N\geq W\). The finite-interval comparison gives \(\kappa {\gt}0\) with \(\kappa \, h_W\leq H^{\mathrm{open}}_{W,2}(A)\), and Lemma 12.10.6 with \(R=2\) yields, for every \(N\geq W\) and every \(v\perp G_N^{\mathrm{ES}}(A)\),
Writing \(\delta =\kappa \gamma _W/(W-1)\), choose \(n\) with \(n\delta {\gt}1\) and put \(m=n+W\); then \(m\delta {\gt}1\) and (591) holds at \(N=m+1\).
Let \(x\mapsto A_x\) be a continuous family of one-site injective tensors of nonzero bond dimension, let \(m\geq 2\), and suppose \(\gamma \lVert v\rVert \leq \lVert H^{\mathrm{open}}_{m+1,2}(A_{x_0})v\rVert \) for \(v\perp G_{m+1}^{\mathrm{ES}}(A_{x_0})\) with \(m\gamma {\gt}1\). Then there is \(\delta {\gt}0\) such that, for all \(x\) near \(x_0\) and all \(N\geq 2m\), the periodic parent Hamiltonian \(H_{N,2}(A_x)\) satisfies \(\delta \lVert v\rVert \leq \lVert H_{N,2}(A_x)v\rVert \) on the orthogonal complement of its kernel.
Choose \(1/m{\lt}\gamma '{\lt}\gamma \). The strict window persists with \(\gamma '\) near \(x_0\), and Knabe’s nearest-neighbor criterion turns it into a periodic gap for every \(N\geq 2m\) with a constant depending only on \(m\) and \(\gamma '\).
If every eigenvalue \(z\) of a Hermitian matrix \(M\) has \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq \delta \), then \(\delta \lVert x\rVert ^2\leq \operatorname {Re}\langle x,Mx\rangle \) for all \(x\perp \ker M\). Conversely, if \(M\geq 0\) satisfies this quadratic-form bound, then every eigenvalue \(z\) of \(M\) has \(\operatorname {Re}z\geq 0\), and \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq \delta \). Let \(0\leq A\leq B\) be matrices with \(\ker A\subseteq \ker B\) and such a gap \(\delta \) for \(A\), and let \(t\geq 0\). Then \(C_t=A+t(B-A)\) is positive semidefinite, \(\ker C_t=\ker A\), and every eigenvalue \(z\) of \(C_t\) satisfies \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq \delta \). If \(\ker A\neq 0\), then \(0\) is an eigenvalue of every \(C_t\).
Since \(t(B-A)\geq 0\), a vector annihilated by \(C_t\) has \(\langle x,Ax\rangle =0\), hence \(Ax=0\), and conversely \(Ax=0\) forces \(Bx=0\). On \((\ker A)^\perp \) one has \(\langle x,C_tx\rangle \geq \langle x,Ax\rangle \geq \delta \lVert x\rVert ^2\). The equivalence with the spectral statement follows by expanding \(x\) in an orthonormal eigenbasis of the Hermitian matrix.
Let \(S\) be a compact subset of a topological space and \(P(\delta ,N,x)\) a property of a real \(\delta \), a natural number \(N\) and a point \(x\) that is preserved by decreasing \(\delta \) and by increasing \(N\). If every \(x\in S\) has \(\delta {\gt}0\) and \(N\) such that \(P(\delta ,N,y)\) holds for all \(y\) near \(x\), then one \(\delta {\gt}0\) and one \(N\) satisfy \(P(\delta ,N,x)\) for all \(x\in S\).
Take a finite subcover of the neighborhoods, the minimum of the finitely many constants and the maximum of the finitely many thresholds.
Let \(E\) be a finite-dimensional complex Hilbert space, let \(x\mapsto H_x\colon E\to E\) be continuous, and suppose the orthogonal projections onto \(\ker H_x\) are continuous in \(x\). For every compact parameter set \(S\), there is \(\delta {\gt}0\) such that \(\delta \lVert v\rVert \leq \lVert H_xv\rVert \) for every \(x\in S\) and \(v\perp \ker H_x\). No positivity or self-adjointness is required.
The restriction of \(H_x\) to \((\ker H_x)^\perp \) is injective, so finite dimensionality gives a positive lower norm bound at each parameter. Half of that bound persists in a neighborhood by continuity of the operator and its kernel projection. A finite subcover of \(S\) gives one positive bound for all parameters.
Let \(x\mapsto A_x\) be a continuous family of one-site injective tensors of nonzero bond dimension, let \(S\) be a compact parameter set, and let \(N\geq 2\). Then there is \(\delta {\gt}0\) such that \(\delta \lVert v\rVert \leq \lVert H_{N,2}(A_x)v\rVert \) for all \(x\in S\) and all \(v\) orthogonal to the kernel of \(H_{N,2}(A_x)\).
The kernel of \(H_{N,2}(A_x)\) is the line spanned by the nonzero periodic matrix product vector of \(A_x\), whose projection is continuous in \(x\). The gap at each point therefore persists in a neighborhood, and a finite subcover of \(S\) gives one constant.
Let \(x\mapsto A_x\) be a continuous family of one-site injective tensors of nonzero bond dimension, and let \(S\) be a compact parameter set. Then there is \(\delta {\gt}0\) such that, for every \(x\in S\), every \(N\geq 2\) and every \(v\perp \ker H_{N,2}(A_x)\),
This is the uniform-gap conclusion of [ 236 , Appendix A, lines 2575–2580 ] for one-site injective tensors and the two-site parent interaction; the source also treats non-injective normal forms [ 176 ] .
Every point of \(S\) has a strict Knabe window, hence a neighborhood with a common gap bound for all \(N\) beyond a threshold. A finite subcover of \(S\) gives one bound and one threshold. The finitely many shorter rings are handled by the previous theorem, and the minimum of these constants works for every \(N\geq 2\).
Let \(x\mapsto A_x\) be continuous with nonzero fixed bond dimension, and let \(S\) be compact. Suppose every \(A_x\) is injective after the same positive length \(p\). For every fixed interaction range \(R{\gt}p\), there is \(\delta {\gt}0\) such that
If each \(A_x\) is merely normal, the same assertion holds at the ranges \(R\geq D^4+1\). No gap estimate or normalized gauge is supplied as a hypothesis.
A pointwise primitive gauge supplies an open-chain gap at a long range. The finite-interval comparison transfers it to range \(p+1\). Choosing a sufficiently long window makes the finite-range Knabe inequality strict. This inequality persists near each parameter, and a finite subcover gives a uniform gap for all sufficiently long rings. The shorter rings have the line generated by their nonzero periodic matrix product vector as kernel. Its projection is continuous, so compactness supplies positive bounds for those finitely many lengths. For a normal tensor, quantum Wielandt gives injectivity at \(D^4\) after a primitive gauge; gauge invariance gives the same length for the original tensor. For a larger prescribed range \(R{\gt}p\), injectivity propagates to length \(R-1\), and the same argument applies.
Let \(x\mapsto (A_x^j)_j\) be a continuous family of blocks. At every length \(L\) at which the length-\(L\) word tuples span the product of the block matrix algebras for every parameter, the projection onto the sum of the local block spaces on \(L\) sites, and the projection onto the local space of the weighted block sum, are continuous in the parameter. Suppose moreover that the physical dimension and all bond dimensions are positive, and that the word tuples span at a length \(S{\gt}0\) for every parameter. For the weighted block sum, the periodic kernel projection is continuous whenever \(S+1\leq R\leq N\). Nonzero block weights may vary arbitrarily. Thus each such fixed volume has a positive gap uniform on any compact parameter set. The overlaps of two local block spaces are also continuous at any common injectivity length.
The joint boundary map has fixed domain, is injective, and has range equal to the sum of the local block spaces. Its range projection therefore depends continuously on the parameter. Restricting the boundary matrices to scalar multiples of the identity gives the injective map onto the periodic ground space. The kernel identity then gives continuity of the periodic kernel projection, after the span at \(S\) is propagated to every length at least \(S\) using the positive dimensions. Compactness yields a common positive finite-volume gap.
Under the preceding simultaneous spanning hypothesis, fix \(R\geq S+1\). On a compact parameter set there is \(\delta {\gt}0\) such that the canonical periodic parent Hamiltonian of the weighted block sum satisfies
In particular, for pairwise inequivalent normalized normal blocks of dimensions \(D_j{\gt}0\), one may choose any \(R\geq 3\max (\sum _jD_j,1)^5\). The simultaneous spanning length is then derived from the dimensions. This includes the two-site interaction of the initially blocked path in [ 236 , Appendix A ] ; see [ 176 ] .
The open-chain multiblock gap gives a strict finite-range Knabe window at each parameter. Continuity preserves the strict inequality in a neighborhood. A finite subcover gives a common gap for all sufficiently long rings. The preceding fixed-volume result handles the finitely many remaining lengths, including \(N=R\). For normalized normal blocks, the simultaneous word-span bound supplies a common length \(S=3\max (\sum _jD_j,1)^5-1\).
Suppose the local ground-space projection and a positive parent interaction \(h_x\) depend continuously on \(x\), with the same local kernel. On a compact parameter set there is \(\kappa {\gt}0\) such that \(h_x\geq \kappa P_{G_R(x)^\perp }\). Consequently every canonical periodic gap uniform in the parameter and chain length transfers to the chosen interactions. This applies, in particular, to compact normal families at every fixed range \(R\geq D^4+1\).
Compactness gives a common positive local gap away from the kernel. Positivity turns it into the stated order bound. Translation and summation preserve this comparison. A pointwise upper comparison identifies the periodic kernels, and the lower comparison transfers the canonical gap with factor \(\kappa \).
Let \((A^j)_j\) have simultaneous spanning length \(S{\gt}0\). An invertible change \(M\) of the physical space preserves this spanning property. More generally, a rectangular change with a left inverse preserves the joint word span. If \(x\mapsto M_x\) is continuous and invertible, then on a compact parameter set the weighted block sums of \((M_xA^j)_j\) have a uniform periodic parent gap at every fixed \(R\geq S+1\), for all \(N\geq R\). All block weights are nonzero.
Expanding each transformed word as a linear combination of original words gives one span inclusion. Applying a left inverse gives the reverse inclusion. Continuity of the transformed letters and the compact-family gap theorem give the assertion.
On the retained physical space let \(Q{\gt}0\) and let \((A^j)_j\) be simultaneously one-site injective. Set \(Q_\gamma =\gamma Q+(1-\gamma )I\) for \(0\leq \gamma \leq 1\). Both the canonical two-site parent Hamiltonians of \((Q_\gamma A^j)_j\) and the periodic Hamiltonians obtained from
where \(h_0\) is the initial canonical parent projection, have a positive gap uniform in \(\gamma \) and every \(N\geq 2\). The positive factor is supplied on the retained physical space. This is the deformation in [ 236 , lines 589–594 and 610–625, Appendix A ] .
The affine path is positive definite, continuous, and invertible. Its endpoints are \(I\) and \(Q\). Joint one-site injectivity persists, so the preceding theorem gives the canonical gap. Inversion and tensor powers are continuous. The conjugated interactions are positive and have exactly the transformed local ground spaces as kernels. Compact positive-interaction comparison then transfers the canonical gap to the displayed interactions.
Let \(P:\mathbb {C}^k\to \mathbb {C}^d\) be a linear map, let \(E\) be the orthogonal projection onto \(\operatorname {ran}P\), and set \(Q=(PP^\dagger )^{1/2}+I-E\). There is a partial isometry \(W\) such that \(Q{\gt}0\), \(P=QW\), and \(WW^\dagger =E\). On the retained physical space this is the polar decomposition of [ 236 , lines 579–584 ] ; the identity extension acts only on the orthogonal complement.
Write \(P^\dagger =VS\) in polar form, with \(V^\dagger V=E\), \(ES=S\), and \(\operatorname {ran}E=\operatorname {ran}S\). The matrix \(S+I-E\) is positive semidefinite. If it annihilates \(x\), multiplying by \(E\) gives \(Sx=0\). Since \(E\) factors through \(S\) and both are Hermitian, \(Ex=0\), and the original equation gives \(x=0\). Thus \(Q=S+I-E\) is positive definite. Set \(W=V^\dagger \). The identities \(VE=V\) and \(VS=P^\dagger \) give \(QW=SV^\dagger =P\). Finally, \(EP=P\) gives \(\operatorname {ran}P\subseteq \operatorname {ran}E\), while \(PV=S\) and \(E=SR\) for some \(R\) give the reverse inclusion.
Let \((A^j)_j\) be pairwise gauge-phase inequivalent normalized normal blocks, with bond dimensions \(D_j\). There is a positive integer \(L\) satisfying \(L+1\leq 3\max (\sum _jD_j,1)^5\) such that blocking \(L\) physical sites gives a simultaneous one-site span of the direct sum of the block matrix algebras. This is the bounded form of the initial blocking in [ 236 , lines 575–597, Appendix A ] , using the bound in [ 13 , lines 317–345 ] .
The simultaneous word-span bound supplies \(L\) and spans the direct sum by length-\(L\) words. The one-site matrices of the blocked blocks are precisely those length-\(L\) word evaluations.
Let \((A^j)_j\) be simultaneously one-site injective blocks of positive bond dimensions. Define the joint physical map by \(P_{i,(j,a,b)}=A^j_i(a,b)\) and write \(P=QW\) as in Theorem 12.10.20. Reading the columns of \(W\) as block matrices \(W^j_i\), set \(A^j_i(\gamma )=\sum _l(Q_\gamma )_{il}W^j_l\), where \(Q_\gamma =\gamma Q+(1-\gamma )I\). This continuous path has endpoints \(W^j\) and \(A^j\), preserves simultaneous spanning at every length, and has coisometric initial physical map on \(\operatorname {ran}P\).
Both its canonical two-site parent Hamiltonians and the positive interactions \(h_\gamma =(Q_\gamma ^{-1})^{\otimes 2} h_0(Q_\gamma ^{-1})^{\otimes 2}\) have a positive periodic gap uniform in \(\gamma \in [0,1]\) and all \(N\geq 2\). Here \(h_0\) is the canonical two-site parent projection of the unweighted block sum of \((W^j)_j\). The interactions are continuous and have exactly the deformed local ground spaces as their kernels.
For pairwise inequivalent normal blocks, there is a positive initial blocking length \(L\) satisfying \(L+1\leq 3\max (\sum _jD_j,1)^5\), after which the constructed path has the canonical uniform gap just stated. The path construction and gap assertion are those of [ 236 , lines 575–625, Appendix A ] .
The left polar factorization gives \(A^j=QW^j\). Since \(Q\) is invertible, the original and polar blocks have the same joint word spans. The affine positive path therefore preserves those spans and has the asserted endpoints. Theorem 12.10.19 gives both uniform gaps and the interaction properties. For normal blocks, the simultaneous span bound supplies \(L\); length-\(L\) evaluations become one-letter evaluations after blocking.
Let \(P:\mathbb {C}^k\to \mathbb {C}^d\), and suppose that unitary matrices \(U\) and \(X\) satisfy \(UP=PX\). Let \(E\) project onto \(\operatorname {ran}P\), and write \(P=QW\) with \(Q=(PP^\dagger )^{1/2}+I-E\) as in Theorem 12.10.20. Then \(U\) commutes with \(PP^\dagger \), \(E\), and \(Q\), and \(UW=WX\). In particular, this applies to the joint physical map of a block family. Physical rotation by a matrix \(L\) replaces that map by \(LP\). These assertions establish the polar step in [ 236 , lines 645–676 ] , with the virtual unitary \(X\) supplied as a hypothesis.
Taking adjoints of \(UP=PX\) and using unitarity gives \(U(PP^\dagger )U^\dagger =PP^\dagger \). The square root and support projection are functions of the positive matrix \(PP^\dagger \), and therefore commute with \(U\). Hence \(Q\) also commutes with \(U\). Since \(Q\) is invertible, the equality \(QUW=UP=PX=QWX\) implies \(UW=WX\). The rotation identity follows by evaluating each physical coordinate.
If \(U\) commutes with a physical matrix \(Q\), then it commutes with \(Q_\gamma =\gamma Q+(1-\gamma )I\) for every \(\gamma \in [0,1]\). In the setting of Theorem 12.10.23, the constructed physical map \(P_\gamma =Q_\gamma W\) therefore obeys \(UP_\gamma =P_\gamma X\) throughout the path. This is the covariance implication of [ 236 , lines 672–676 ] ; it does not assert the existence of a virtual unitary from an arbitrary state symmetry.
Commutation is preserved under scalar multiplication and addition. Thus \(UP_\gamma =UQ_\gamma W=Q_\gamma UW=Q_\gamma WX=P_\gamma X\).
Let \(A\) be a tensor, let \(U,Q\) be physical matrices with \(UQ=QU\), and write \(T_L(V)=V^{\otimes L}\). Then \(T_L(U)\) and \(T_L(Q)\) commute. If \(T_L(U)G_L(A)=G_L(A)\), then \(T_L(U)G_L(QA)=G_L(QA)\). When \(Q\) is invertible, these two invariance statements are equivalent. If \(U\) is unitary and \(G_L(A)\) is invariant, its canonical complementary projection \(h_L(A)\) commutes with \(T_L(U)\). Finally, if \(Q\) is invertible and Hermitian, every interaction \(h\) commuting with \(T_L(U)\) retains this symmetry under the deformation \(T_L(Q^{-1})hT_L(Q^{-1})\).
Tensor powers preserve matrix products, and \(G_L(QA)=T_L(Q)G_L(A)\). Applying the commuting operators in either order gives the forward invariance statement. Applying the same argument with \(Q^{-1}\) gives the converse. A unitary preserving a subspace also preserves its orthogonal complement and commutes with the two orthogonal projections. The inverse of \(Q\) commutes with \(U\), so the final assertion follows by multiplication.
Let \(h\) be an interaction on \(R\) sites, let \(U\) be a physical matrix, and suppose \([h,U^{\otimes R}]=0\). For every \(N\geq R\), each cyclic extension of \(h\) commutes with \(U^{\otimes N}\). Consequently both the periodic sum and the sum over nonwrapping windows commute with \(U^{\otimes N}\). In configuration coordinates, cyclic extension is the matrix obtained by placing \(h\) on the window and the identity on its complement.
Reordering the physical sites into the window and its complement identifies the extended interaction with \(h\otimes I\) and the global action with \(U^{\otimes R}\otimes U^{\otimes (N-R)}\). These commute by the hypothesis. Summing over all cyclic windows, or only the nonwrapping ones, preserves commutation.
Let \(A=(A^j)_j\) be a finite block family, assembled with unit coefficients, and let \(P=QW\) be its constructed joint left polar decomposition. Suppose that \(U\) is unitary, \(UQ=QU\), and \(U^{\otimes 2}G_2(A)=G_2(A)\). For every \(\gamma \in [0,1]\), put \(Q_\gamma =\gamma Q+(1-\gamma )I\) and \(P_\gamma =Q_\gamma W\). Then \(G_2(W)\) and \(G_2(P_\gamma )\) are invariant under \(U^{\otimes 2}\). Both the canonical parent projection of \(P_\gamma \) and the positive interaction
commute with \(U^{\otimes 2}\). Their open and periodic Hamiltonians commute with \(U^{\otimes N}\) for every \(N\geq 2\). More generally, supplied invariance of \(G_L(A)\) implies invariance of \(G_L(W)\) and \(G_L(P_\gamma )\) for any fixed \(L\). This is the conditional parent-symmetry implication of [ 236 , lines 672–676 ] . The original local-space invariance and commutation of \(Q\) are explicit hypotheses here. Theorem 12.10.37 derives both from symmetry up to a phase of jointly injective blocks in the source’s standard form.
The positive factor \(Q\) is invertible, and \(A=QW\). Invariance therefore passes to \(W\) by the inverse deformation. Each \(Q_\gamma \) commutes with \(U\), so invariance passes from \(W\) to \(P_\gamma \). The invariant local spaces give commuting canonical projections, and inverse conjugation preserves this commutation. The open and periodic conclusions follow by extending and summing the local interactions.
Let \(A\) be a one-site injective tensor of positive bond dimension \(D\), in the unital canonical orientation \(\mathcal{E}_A(I)=I\). Let \(g\mapsto U_g\) be a unitary representation of a monoid \(G\), and suppose that the matrix product vectors of \(A\) are unchanged by \(U_g^{\otimes N}\) for every \(g\) and every chain length \(N\). For each \(g\), there is a virtual unitary \(V_g\) such that \(\sum _l(U_g)_{il}A_l=V_gA_iV_g^\dagger \) for every \(i\).
For the single-block physical map \(P_{i,(a,b)}=A_i(a,b)\), the induced unitary \(X_g=V_g^T\otimes V_g^\dagger \) satisfies \(U_gP=PX_g\). Consequently, \(U_g\) commutes with the extended positive polar factor \(Q=(PP^\dagger )^{1/2}+I-E\), and the same \(X_g\) satisfies \(U_gP_\gamma =P_\gamma X_g\) at every \(\gamma \in [0,1]\). No faithful adjoint fixed point or virtual covariance is supplied as an additional hypothesis.
With the displayed coordinates for \(P\), unital normalization means \(\sum _iA_iA_i^\dagger =I\). The condition \(\operatorname {tr}_{\mathrm{left}}(P^\dagger P)=I\) printed in [ 236 , line 653 ] instead means \(\sum _iA_i^\dagger A_i=I\); these are the two dual canonical orientations. This result treats the already-blocked, single-block unital orientation of the argument in [ 236 , lines 645–676 ] , for exact invariance of the matrix product vectors. Several blocks are treated in Theorem 12.10.36.
Injectivity implies irreducibility. The adjoint transfer map is a trace-preserving irreducible quantum channel, and therefore has a faithful invariant density matrix. The fundamental theorem gives an invertible gauge for each symmetric twist. In the stated normalization this gauge is a nonzero scalar multiple of a virtual unitary, by Theorem 10.4.10.2; the scalar cancels in conjugation. Taking the induced action on matrix entries gives \(X_g\), which is unitary as a tensor product of unitaries. The polar covariance and affine covariance theorems give the remaining conclusions, including the identity extension off the physical support.
Let \(A\) be one-site injective, of positive bond dimension, with \(\sum _iA_i^\dagger A_i=I\). Let \(g\mapsto U_g\) be a unitary representation of a monoid, and suppose that \(U_g^{\otimes N}\) fixes the matrix product vector of \(A\) for every \(g\) and \(N\). For each \(g\) there is a virtual unitary \(V_g\) such that \(\sum _l(U_g)_{il}A_l=V_gA_iV_g^\dagger \) for every \(i\). For \(P_{i,(a,b)}=A_i(a,b)\), the induced unitary \(X_g=V_g^T\otimes V_g^\dagger \) satisfies \(U_gP=PX_g\). Hence \(U_g\) commutes with the identity-extended positive polar factor \(Q\), and the same \(X_g\) intertwines \(U_g\) with every \(P_\gamma =Q_\gamma W\) in the constructed deformation.
The normalization here is exactly \(\operatorname {tr}_{\mathrm{left}}(P^\dagger P)=I\) in [ 236 , line 653 ] . These conclusions cover the already-blocked single injective block with exact vector symmetry. Symmetry up to a phase, for one or several blocks, is treated in Theorem 12.10.36; see [ 160 ] .
The fundamental theorem gives an invertible virtual gauge with phase one. Injectivity implies irreducibility, while unitary physical rotation preserves irreducibility and the trace-preserving normalization. The unitary replacement lemma therefore gives a unitary virtual gauge with the same phase one. Its action on the matrix-entry pairs is \(X_g\), a tensor product of unitaries. The polar covariance theorem gives \([U_g,Q]=0\), including the identity extension off support, and the affine covariance theorem preserves the same intertwiner throughout the path.
Let \(A\) be a one-site injective tensor of positive bond dimension, in the unital canonical orientation \(\mathcal{E}_A(I)=I\). Let \(g\mapsto U_g\) be a unitary representation of a monoid, and suppose that \(U_g^{\otimes N}\) fixes the matrix product vector of \(A\) for every \(g\) and \(N\). Then \(U_g^{\otimes L}G_L(A)=G_L(A)\) for every \(L\). Along the constructed single-block polar deformation \(P_\gamma =Q_\gamma W\), both the canonical two-site parent projection and the positive interaction
commute with \(U_g^{\otimes 2}\). Their open and periodic sums commute with \(U_g^{\otimes N}\) for every \(N\geq 2\). The initial local-space invariance follows already from injectivity and exact state symmetry, without the normalization assumption. This proves the parent-symmetry conclusion of [ 236 , lines 645–676 ] for the stated single-block unital case, dual to the printed partial-trace normalization as explained in Theorem 12.10.28. Several blocks are treated in Theorem 12.10.37.
The fundamental theorem gives gauge equivalence between \(A\) and its physical twist by each \(U_g\). Gauge equivalence preserves every local MPS space, so these spaces are invariant under the physical action. The normalized single-block covariance theorem gives \([U_g,Q]=0\). The supplied-invariance parent-symmetry theorem now applies, proving the local and global conclusions throughout the deformation.
Let \(A\) be a one-site injective tensor of positive bond dimension, with \(\sum _i A_i^\dagger A_i=I\). Let \(g\mapsto U_g\) be a unitary representation of a monoid, and suppose that \(U_g^{\otimes N}\) fixes the matrix product vector of \(A\) for every \(g\) and \(N\). For its constructed single-block polar deformation \(P_\gamma =Q_\gamma W\), both the canonical two-site parent projection and the positive interaction
commute with \(U_g^{\otimes 2}\). Their open and periodic sums commute with \(U_g^{\otimes N}\) for every \(N\geq 2\). In the coordinates \(P_{i,(a,b)}=A_i(a,b)\), the normalization is precisely \(\operatorname {tr}_{\mathrm{left}}(P^\dagger P)=I\). This is the already-blocked single-block parent-symmetry conclusion of [ 236 , lines 645–676 ] in the printed canonical orientation. Several blocks are treated in Theorem 12.10.37.
The fundamental theorem gives gauge equivalence between \(A\) and its physical twist by \(U_g\), hence \(U_g^{\otimes L}G_L(A)=G_L(A)\) at every length \(L\). The trace-preserving covariance theorem gives \([U_g,Q]=0\). Applying the supplied-invariance parent-symmetry theorem proves commutation of both local interactions and their open and periodic sums.
Let \((A^k)_k\) and \((B^k)_k\) be two families of blocks of the same bond dimensions \(D_k\), each of whose one-site tuples \((A^k_i)_k\) and \((B^k_i)_k\) span \(\bigoplus _kM_{D_k}(\mathbb {C})\). Suppose that \(\sum _k\operatorname{tr}A^k_w=\sum _k\operatorname{tr}B^k_w\) for every word \(w\), where \(A^k_w\) denotes the ordered product along \(w\). Then there is an algebra automorphism \(T\) of \(\bigoplus _kM_{D_k}(\mathbb {C})\) with \(T\bigl((A^k_w)_k\bigr)=(B^k_w)_k\) for every word \(w\).
Let \(c\) be a finite linear combination of words with \(\sum _wc_w(A^k_w)_k=0\), and put \(\Delta =\sum _wc_w(B^k_w)_k\). For every letter \(i\), equality of the word traces gives \(\sum _k\operatorname{tr}(\Delta _kB^k_i)=\sum _wc_w\sum _k\operatorname{tr}B^k_{wi}\), and this equals \(\sum _wc_w\sum _k\operatorname{tr}A^k_{wi}=0\). The one-site tuples of the second family span the direct sum, and the trace pairing on it is nondegenerate, so \(\Delta =0\). Hence the assignment \((A^k_w)_k\mapsto (B^k_w)_k\) extends to a well-defined linear map \(T\) on the span of the words of the first family, which is the whole direct sum. The same argument with the families exchanged shows that \(T\) is injective; it is surjective because the words of the second family also span. It sends the empty word to the identity and concatenation to products, so it is multiplicative on words and, by bilinearity, on the whole algebra.
Let \(g\mapsto U_g\) be a linear representation of a monoid on the physical space. A tensor \(A\) is on-site symmetric up to a phase if for every \(g\) and every chain length \(N\) there is a scalar \(\lambda _{g,N}\) with \(U_g^{\otimes N}|\psi _N\rangle =\lambda _{g,N}|\psi _N\rangle \), where \(|\psi _N\rangle \) is the matrix product vector of \(A\) of length \(N\). The scalars may depend on both \(g\) and \(N\). Exact on-site symmetry is the case \(\lambda _{g,N}=1\). This is the symmetry of the state in [ 236 , lines 655–657, 2590–2593 ] .
Let \(\phi \) be a linear functional on a complex algebra \(\mathcal A\) with \(\phi (1)\neq 0\) such that \(\phi (MN)=0\) for all \(N\) forces \(M=0\), and let \((x_i)_i\) and \((y_i)_i\) be finite families spanning \(\mathcal A\). If \(\phi (y_w)=c_n\phi (x_w)\) for every word \(w\) of length \(n\in \{ 1,2,3\} \), where \(x_w\) and \(y_w\) denote the ordered products along \(w\), then \(\phi (y_w)=c_1^{|w|}\phi (x_w)\) for every word \(w\).
Consequently, let \(A^1,\dots ,A^r\) be blocks of positive bond dimensions whose one-site tuples span \(\bigoplus _kM_{D_k}(\mathbb {C})\), let \(g\mapsto U_g\) be a unitary representation of a monoid, and suppose that for every \(g\) and \(N\) there is a scalar \(\lambda _{g,N}\) with \(U_g^{\otimes N}|\psi _N\rangle =\lambda _{g,N}|\psi _N\rangle \) for the matrix product vectors \(|\psi _N\rangle \) of \(\bigoplus _kA^k\). Then for each \(g\) there is \(c\) with \(|c|=1\) such that \(\sum _k\operatorname{tr}B^k_w=c^{|w|}\sum _k\operatorname{tr}A^k_w\) for every word \(w\), where \(B^k_i=\sum _l(U_g)_{il}A^k_l\). Exact symmetry is the case \(\lambda _{g,N}=1\). A symmetry of the state holds only up to such a phase [ 236 , lines 1313–1314, 2590–2593 ] .
Nondegeneracy against the spanning family \((y_i)\) makes \(x_i\mapsto y_i\) a well-defined linear map \(L\): if \(\sum _ia_ix_i=0\), then \(\phi \bigl((\sum _ia_iy_i)y_j\bigr)=c_2\phi \bigl((\sum _ia_ix_i)x_j\bigr)=0\) for every \(j\). By multilinearity, \(\phi (LX)=c_1\phi (X)\), \(\phi (LX\, LY)=c_2\phi (XY)\) and \(\phi (LX\, LY\, LZ)=c_3\phi (XYZ)\) for all \(X,Y,Z\). The map \(L\) is onto, so pairing against \(LZ\) gives \(LX\, LY=L(XY)L(1)\) and \(c_1L(1)=c_2\cdot 1\). Since \(\phi (1)\neq 0\) lies in the image of \(\phi \circ L=c_1\phi \), we have \(c_1\neq 0\); applying \(\phi \) to \(c_1L(1)=c_2\cdot 1\) gives \(c_2=c_1^2\), hence \(L(1)=c_1\cdot 1\) and \(LX\, LY=c_1L(XY)\). Induction on the word gives \(y_w\, L(1)=c_1^{|w|}L(x_w)\), and applying \(\phi \) proves the first claim.
For the blocks take \(\phi (M)=\sum _k\operatorname{tr}M_k\), \(x_i=(A^k_i)_k\) and \(y_i=(B^k_i)_k\); the latter span because \(U_g\) is invertible, and the trace pairing on \(\bigoplus _kM_{D_k}(\mathbb {C})\) is nondegenerate with \(\phi (1)=\sum _kD_k\neq 0\) (if \(r=0\) there is nothing to prove). The hypothesis at lengths one, two and three gives the three relations, so \(c=\lambda _{g,1}\) works. At length one the nonzero vector \(v_i=\phi (x_i)\) satisfies \(U_gv=cv\), and unitarity of \(U_g\) gives \(|c|=1\).
Let \(A^1,\dots ,A^r\) be blocks of positive bond dimensions \(D_k\) whose one-site tuples \((A^k_i)_k\) span \(\bigoplus _kM_{D_k}(\mathbb {C})\). Let \(g\mapsto U_g\) be a unitary representation of a monoid on the physical space, and suppose that \(U_g^{\otimes N}\) maps every matrix product vector \(|\psi _N\rangle \) of the block-diagonal tensor \(\bigoplus _kA^k\) to a scalar multiple \(\lambda _{g,N}|\psi _N\rangle \). Then for each \(g\) there are a scalar \(c\) with \(|c|=1\), a permutation \(\sigma \) of the blocks with \(D_{\sigma (k)}=D_k\) and invertible matrices \(X_k\) such that
for every block \(k\) and letter \(i\). If in addition every block satisfies \(\sum _iA^{k\dagger }_iA^k_i=I\), the matrices \(X_k\) may be taken unitary. The joint one-site span is the standard form of [ 236 , lines 325–352 ] . This is the several-block form of the virtual action in [ 236 , lines 652–660 ] .
By Lemma 12.10.34 there is \(c\) with \(|c|=1\) such that the summed block traces of the blocks twisted by \(U_g\) are \(c^{|w|}\) times those of \((A^k)_k\). Hence the summed block traces of \((A^k)_k\) and of the blocks twisted by the unitary \(\bar cU_g\) agree on every word. These twisted blocks still span jointly at one site, because \(\bar cU_g\) is invertible. Lemma 12.10.32 gives an algebra automorphism sending the one-site tuples of the original blocks to those of the twisted blocks, and Theorem 23.8.1.3 decomposes it as a block permutation preceded by blockwise conjugations. In the trace-preserving normalization each block is injective, hence irreducible, and unitary physical rotation preserves both irreducibility and the normalization. The unitary replacement lemma then makes each conjugating matrix unitary. Multiplying by \(c\) returns to the twist by \(U_g\).
In the setting of Theorem 12.10.35, assume \(\sum _iA^{k\dagger }_iA^k_i=I\) for every block. For the joint physical map \(P_{i,(k,a,b)}=A^k_i(a,b)\) there is a unitary \(X_g\) on \(\bigoplus _k\mathbb {C}^{D_k}\otimes \mathbb {C}^{D_k}\) with \(U_gP=PX_g\). Consequently \(U_g\) commutes with the identity-extended positive polar factor \(Q\), and the same \(X_g\) satisfies \(U_gP_\gamma =P_\gamma X_g\) along the constructed deformation \(P_\gamma =Q_\gamma W\). The block-diagonal normalization is exactly \(\operatorname {tr}_{\mathrm{left}}(P^\dagger P)=I\) of [ 236 , line 653 ] , and the conclusion is the covariance \(U_gP=PX_g\) of [ 236 , lines 652–660 ] for several blocks, under symmetry of the matrix product vectors up to a phase; see [ 160 ] .
Let \(c\), \(\sigma \) and the unitaries \(V_k\) be as in Theorem 12.10.35. Define \(X'_g\) by sending the matrix coordinates of block \(k\) through \(V_k^T\otimes V_k^\dagger \) and then to the coordinates of block \(\sigma (k)\). It is the product of a block-diagonal unitary and a permutation of coordinates, hence unitary. Column \((\sigma (k),a,b)\) of \(\bar cU_gP\) is the \((a,b)\) entry of block \(\sigma (k)\) twisted by \(\bar cU_g\), which equals \((V_kA^k_iV_k^\dagger )(a,b)\); this is the corresponding column of \(PX'_g\). Hence \(X_g=cX'_g\) is unitary and satisfies \(U_gP=PX_g\). The polar and affine covariance theorems give the remaining conclusions.
In the setting of Theorem 12.10.35, \(U_g^{\otimes L}G_L(\bigoplus _kA^k)=G_L(\bigoplus _kA^k)\) for every \(L\). If every block satisfies \(\sum _iA^{k\dagger }_iA^k_i=I\), then along the constructed deformation \(P_\gamma =Q_\gamma W\) both the canonical two-site parent projection and the positive interaction
commute with \(U_g^{\otimes 2}\), and their open and periodic sums commute with \(U_g^{\otimes N}\) for every \(N\geq 2\). This is the parent-symmetry conclusion of [ 236 , lines 672–676 ] for several blocks in the printed standard form, for symmetry of the matrix product vectors up to a phase.
The local MPS space of the block-diagonal tensor is the sum of the local spaces of its blocks. The physical action carries it to the sum of the local spaces of the twisted blocks. By Theorem 12.10.35, the twist of block \(\sigma (k)\) by \(\bar cU_g\) is gauge equivalent to block \(k\), and gauge equivalence preserves local spaces, so the permuted sum is the original one. The action of \(U_g^{\otimes L}\) is \(c^L\neq 0\) times that of \((\bar cU_g)^{\otimes L}\) and has the same image. Together with \([U_g,Q]=0\) from Theorem 12.10.36, the supplied-invariance parent-symmetry theorem gives the remaining conclusions.
Let \((A^j)_j\) be simultaneously one-site injective blocks of positive bond dimensions, let all block weights be nonzero, and let \(Q{\gt}0\) on the retained physical space. Set \(Q_\gamma =\gamma Q+(1-\gamma )I\) for \(\gamma \in [0,1]\). The canonical two-site open parents of \((Q_\gamma A^j)_j\) have a positive gap uniform in \(\gamma \) and every length \(N\geq 2\). The same conclusion holds for the open Hamiltonians with interaction \(h_\gamma =(Q_\gamma ^{-1})^{\otimes 2}h_0(Q_\gamma ^{-1})^{\otimes 2}\), where \(h_0\) is the initial canonical parent projection. In both assertions the gap is on the orthogonal complement of the open Hamiltonian’s kernel. This is the open-chain consequence of the deformation in [ 236 , lines 589–594 and 610–625, Appendix A ] .
The path \(Q_\gamma \) is continuous and invertible, so joint one-site spanning persists. The compact open-chain theorem at range two gives the canonical gap. The displayed positive interaction varies continuously and has exactly the transformed two-site ground space as its kernel. The compact positive-interaction theorem gives its uniform open gap.
Let \((A^j)_j\) be simultaneously one-site injective blocks of positive bond dimensions, and form their joint polar decomposition \(P=QW\). Along the constructed path \(A^j(\gamma )=Q_\gamma W^j\), both the canonical two-site open parents and the open Hamiltonians with interaction \(h_\gamma =(Q_\gamma ^{-1})^{\otimes 2} h_0(Q_\gamma ^{-1})^{\otimes 2}\) have a positive gap uniform in \(\gamma \in [0,1]\) and every \(N\geq 2\); here \(h_0\) is the canonical two-site parent projection of the unweighted sum of the polar blocks.
For pairwise inequivalent normalized normal blocks, each gap assertion applies after finite initial blocking at a positive length \(L\) with \(L+1\leq 3\max (\sum _jD_j,1)^5\). The chain length then counts blocked sites. The path is that of [ 236 , lines 575–625, Appendix A ] .
Invertibility of \(Q\) transfers the joint one-site span from \(A^j\) to \(W^j\). Apply Theorem 12.10.38 to these polar blocks and the constructed positive factor. For normal blocks, the simultaneous spanning bound supplies \(L\); length-\(L\) words become one-letter matrices after blocking.
Let \(S\) be a compact subset of a topological parameter space. For each \(N\in \mathbb N\), let \(E_N,F_N\) be seminormed additive groups, and let \(H_N(x):E_N\to F_N\) and \(V_N(x)\subseteq E_N\) be families of maps and admissible vectors. Suppose that every \(x\in S\) has a neighborhood \(U_x\), a constant \(\delta _x{\gt}0\), and a threshold \(N_x\) such that \(\delta _x\| v\| \leq \| H_N(y)v\| \) whenever \(y\in U_x\), \(N\geq N_x\), and \(v\in V_N(y)\). Then there are \(\delta {\gt}0\) and \(N_0\) such that \(\delta \| v\| \leq \| H_N(x)v\| \) for every \(x\in S\), \(N\geq N_0\), and \(v\in V_N(x)\). Linearity of the maps is unnecessary. This is the norm-bound form of the compactness argument in [ 236 , Appendix A ] .
Choose a finite subcover of the neighborhoods \(U_x\). Let \(\delta \) be the minimum of the corresponding positive constants and \(N_0\) the maximum of their thresholds. Decreasing the constant preserves the inequality because \(\| v\| \geq 0\), and increasing the threshold restricts the lengths under consideration.
12.10.1 Additional joint boundary continuity and compactness inputs
Let \(A^j\) be a finite family of tensors with fixed physical dimension and virtual dimensions \(D_j\). On the diagonal virtual boundary space define
Equip the domain with the direct sum of the Hilbert–Schmidt inner products and the codomain with the usual configuration inner product. Its range is \(\sum _jG_L(A^j)\).
Every complex linear functional on \(\prod _j\operatorname {Mat}_{D_j}(\mathbb C)\) has the form \(M\mapsto \sum _j\operatorname{tr}(\Delta _jM_j)\) for some family \(\Delta \). Consequently, a linear subspace \(W\) is the whole product algebra if the only family satisfying \(\sum _j\operatorname{tr}(\Delta _jM_j)=0\) for every \(M\in W\) is the zero family.
The matrix trace pairing is nondegenerate. Apply its isomorphism with the dual space to the restriction of the functional to each summand. A proper subspace has a nonzero linear functional vanishing on it, so the asserted separation condition excludes every proper subspace.
The tuples of length-\(L\) word products span \(\prod _j\operatorname {Mat}_{D_j}(\mathbb C)\) if and only if \(\Gamma _L^{\oplus }\) is injective.
If \(\Gamma _L^{\oplus }((X_j)_j)=0\), the trace pairing with \((X_j)_j\) vanishes on every word tuple and hence on the entire product algebra. Nondegeneracy of the matrix trace pairing gives \(X_j=0\) for every \(j\). Conversely, injectivity implies that the only trace pairing vanishing on the word span is zero. The preceding duality lemma then gives the full product span.
Let \(x\mapsto A_x^j\) be continuous for every block \(j\), with fixed physical and virtual dimensions, and suppose that the joint boundary map at length \(L\) is injective for every parameter \(x\). For arbitrary nowhere-zero weights \(\mu _j(x)\), set \(B_x=\bigoplus _j\mu _j(x)A_x^j\). Then the projector onto \(G_L(B_x)\) and the canonical interaction \(h_L(B_x)\) depend continuously on \(x\). At every fixed volume \(N\geq L\), the corresponding open and periodic parent Hamiltonians are continuous as well. Continuity of the weights is unnecessary. These are the finite-window continuity inputs in [ 236 , Appendix A, lines 2575–2578 ] .
Each entry of the joint boundary map is a sum of traces of finite products of continuous tensor letters. The joint map is injective, so its range projection is continuous by the inverse-Gram formula. Nonzero weights do not change the sum of the block ground spaces. The canonical interaction is the complementary projection, and the finite-volume Hamiltonians are finite sums of its fixed embeddings.
For a continuous family of fixed-dimensional blocks, the parameters where \(\Gamma _L^{\oplus }\) is injective form an open set, as do the parameters where the length-\(L\) word tuples span the full product matrix algebra. If \(L{\gt}0\), either condition implies the corresponding condition at every length \(n\geq L\). No normalization is required.
Injectivity of a bounded map with finite-dimensional domain is open in operator norm, and trace duality identifies injectivity with simultaneous spanning. For propagation, a boundary \((X_j)_j\) giving zero at length \(L+1\) also gives \(\Gamma _L^{\oplus }((A^{j,i}X_j)_j)=0\) for every letter \(i\). Injectivity at \(L\) gives \(A^{j,i}X_j=0\). Every nonempty word therefore annihilates \(X_j\), so \(\Gamma _L^{\oplus }((X_j)_j)=0\) because \(L{\gt}0\). Injectivity gives \(X_j=0\). Iterate this argument.
Let \(S\) be compact and let \(x\mapsto A_x^j\) be continuous, with fixed physical and block dimensions. Suppose that for every \(x\in S\) there is a positive length \(L_x\) at which the simultaneous word products span the full product matrix algebra. Then there is one \(L{\gt}0\) such that simultaneous spanning holds for every \(x\in S\) and every length \(n\geq L\). No normalization or positivity of the block dimensions is required.
Spanning at \(L_x\) persists in a neighborhood of \(x\) by openness and at all larger lengths by positive-length propagation. A finite subcover of \(S\) gives a common positive length by taking the maximum of its finitely many thresholds. If \(S\) is empty, any positive length suffices. This supplies the common finite-window choice used in [ 236 , Appendix A, lines 2499–2503 and 2575–2578 ] .
Let \(S\) be compact and let \(x\mapsto A_x^j\) be continuous, with fixed block dimensions. Suppose that every block is trace preserving on \(S\) and that, for each \(x\in S\), the simultaneous word products span the full product matrix algebra at some length. Then there is one \(L\) such that \(\Gamma _n^{\oplus }(A_x)\) is injective for all \(x\in S\) and all \(n\geq L\). In particular, the spanning hypothesis holds for normalized primitive blocks with positive virtual dimensions and positive definite fixed points that are pairwise inequivalent under similarity and phase. No continuity of the fixed-point matrices is required.
Trace preservation extends simultaneous spanning by one letter, including when the initial length is zero. Apply the preceding theorem to these positive lengths, and use trace duality to obtain joint-boundary injectivity. For primitive blocks, simultaneous spanning follows from inequivalence of the blocks.
Suppose that the orthogonal projection onto \(G_L(A_x)\) is continuous in \(x\). Then its complementary canonical interaction is continuous. For each fixed \(N\geq L\), every translated local term and the open and periodic parent Hamiltonians are continuous as well.
Subtract the local projection from the identity. Its fixed embeddings into the chain are continuous, and so are their finite sums.
Let \(x\mapsto A_x^j\) be continuous and injective after blocking \(L\) sites for every \(x\) and \(j\). For each pair of blocks \(i,j\), the norm \(\| P_{G_L(A_x^i)}P_{G_L(A_x^j)}\| \) is continuous in \(x\).
Both projections are continuous in operator norm, as are their product and its norm.
Let \(x\mapsto A_x^j\) be continuous, with fixed block dimensions, and suppose that their length-\(L\) word tuples span the product matrix algebra for every \(x\). Then the projection onto \(\sum _jG_L(A_x^j)\) is continuous. The same holds for the ground space of any weighted block sum with nowhere-zero weights.
Simultaneous word spanning implies injectivity of the joint boundary map, so the continuity theorem applies.