14 Commuting Parent Hamiltonians and Spectral Gaps
For normal matrix product states, the first sections compare commutation of neighboring parent terms with renormalization fixed points and zero correlation length under the stated hypotheses. Decorrelation is expressed through commuting support projectors and their idempotent product.
The remaining sections establish spectral-gap estimates for frustration-free sums of local projections. The anticommutator form of the martingale criterion and cyclic norm-compression bounds turn control of overlapping ground spaces into a uniform lower bound above the ground space.
14.1 Commuting Local Terms for Parent Hamiltonians
This section concerns the commutation equation for local terms in a periodic chain. For normal tensors it relates renormalization fixed points to the tensors whose length-two local terms commute, \(h_i(A,2)h_j(A,2)=h_j(A,2)h_i(A,2)\); the implication from commuting terms back to a renormalization fixed point uses a left-canonical normalization.
The nearest-neighbor commuting parent Hamiltonian condition in [ CPGSV16 , Definition 3.9 and Theorem 3.10 ] is distinct from the parent commuting Hamiltonian condition of [ CPGSV16 , Appendix D.2 ] , which is stated for local projectors \(Q_{AX}\) and \(Q_{XB}\). The latter appears in the decorrelation section that follows. In [ CPGSV16 , Definition 3.9 ] the parent-Hamiltonian condition also says that, for \(N{\gt}L\), the ground space is spanned by the periodic MPS vectors associated to the BNT components; nearest-neighbor means the specialization \(L=2\).
The proof of the forward implication in [ CPGSV16 , Theorem 3.10 ] is the single sentence at source lines 1305–1307 saying that it follows immediately from the structural characterization in Theorem 3.11. It does not invoke Definition D.2. The three-site kernel intersection proved below is instead a separate local specialization of that Appendix D definition.
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 [ CPGSV16 , 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 [ CPGSV16 , 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 [ CPGSV16 , 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 two supports appearing in [ CPGSV16 , Definition D.2 ] ; it does not assert that they commute.
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 14.1.1. 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 [ CPGSV16 , Appendix D.2 ] projectors associated with the [ CPGSV16 , Appendix B ] basic-vector form is a separate step.
By Theorem 14.1.3, 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 [ CPGSV16 , 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 14.1.3 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 14.1.1 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 [ CPGSV16 , 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 [ CPGSV16 , 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 (??).
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, and the family \(p_i=h_i(A,2)\) satisfies the finite-chain local-projector hypotheses used in the conditional Appendix B extraction.
By Lemma 14.1.11, overlapping length-two cyclic supports force the three alternatives in (??). The equal-start case gives \(h_i(A,2)=h_j(A,2)\). The two nearest-neighbor cases are (??) and its transpose. The overlap-reduction lemma supplies the remaining disjoint pairs by locality.
Let \(N\ge 2\). Suppose that the translated length-two local terms commute for every pair of overlapping cyclic windows: \(h_i(A,2)h_j(A,2) = h_j(A,2)h_i(A,2)\). Then the family \(p_i=h_i(A,2)\) gives the finite-chain local-projector hypotheses used in the conditional Appendix B extraction.
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 [ CPGSV16 , 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 14.1.15.
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 [ CPGSV16 , 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 [ CPGSV16 , Theorem 3.10(iii) ] quantifies this condition over all \(N{\gt}2\).
This is the source quantification in [ CPGSV16 , Theorem 3.10(iii) ] . For every \(N{\gt}2\), the fixed-chain condition of Definition 14.1.16 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 above contains this commuting equation together with the frustration-free equation for \(V^{(N)}(A)\); it does not assert the source ground-space spanning property. For \(N\ge 2\), Lemma 13.3.19 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 14.1.17. 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 is the easy inclusion in the source ground-space spanning equation of [ CPGSV16 , Definition 3.9 ] ; the reverse inclusion is the remaining spanning direction.
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 14.3.12 gives \(\ker H_L^{(N)}(B)\subseteq \mathcal G_{N,L}(B)\). Together with the splitting hypothesis,
The blockwise containments give
and Lemma 15.1.4 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 14.1.17 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 easy inclusion above gives
and the assumed reverse containment gives equality. For \(L=2\), the all-chain nearest-neighbor condition of Definition 14.1.17 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, the easy inclusion of Lemma 14.1.20 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 [ CPGSV16 ] , 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 [ CPGSV16 ] , 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 is the usual parent-Hamiltonian frustration-freeness statement. The separate ground-space spanning clause of the source definition is not asserted here.
Let \(A\) have positive bond dimension and be a normal renormalization fixed-point tensor in the sense of [ CPGSV16 ] . 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.
This statement is restricted to one normal sector. It does not assert the corresponding result for canonical-form tensors with several sectors or repeated copies.
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 14.1.28 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.
This statement treats one copy of every distinct sector. It does not include repeated copies or arbitrary raw sector weights.
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
where the latter identities follow 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 is the multiplicity-one distinct-sector case of [ CPGSV16 , Theorem 3.10 ] . It does not include repeated copies of a basis tensor or arbitrary raw sector weights.
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. This is the multiplicity-one distinct-sector forward implication in [ CPGSV16 , Definition 3.9 and Theorem 3.10 ] . It does not include repeated copies or arbitrary raw sector weights.
Theorem 14.1.30 gives pairwise commutation of the translated two-site interactions for every \(N{\gt}2\). The parent-Hamiltonian construction gives the zero-energy equation. Theorem 14.1.31 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 14.1.34. 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 [ CPGSV16 , 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 [ CPGSV16 , 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{tenkzfree}
\tnput[boundary, ports={east:virtual}]
{bondLeft}{(-12mm,0)}{}
\tnput[box, ports={west:virtual,east:virtual,north:physical}]
{bondUzero}{(0,0)}{U_0}
\tnput[dot, ports={west:virtual,east:virtual},
label pos=south]
{bondPhiZero}{(16mm,0)}{\varphi_{01}}
\tnput[box, ports={west:virtual,east:virtual,north:physical}]
{bondUone}{(32mm,0)}{U_1}
\tnput[dot, ports={west:virtual,east:virtual},
label pos=south]
{bondPhiOne}{(48mm,0)}{\varphi_{12}}
\tnput[box, ports={west:virtual,east:virtual,north:physical}]
{bondUtwo}{(64mm,0)}{U_2}
\tnput[boundary, ports={west:virtual}]
{bondRight}{(76mm,0)}{}
\tnput[boundary, ports={south:physical}]
{bondPhysicalZero}{(0,14mm)}{}
\tnput[boundary, ports={south:physical}]
{bondPhysicalOne}{(32mm,14mm)}{}
\tnput[boundary, ports={south:physical}]
{bondPhysicalTwo}{(64mm,14mm)}{}
\tnjoin[name=bondOuterLeft]{bondLeft.east}{bondUzero.west}
\tnjoin[name=bondZeroLeft]{bondUzero.east}{bondPhiZero.west}
\tnjoin[name=bondZeroRight]{bondPhiZero.east}{bondUone.west}
\tnjoin[name=bondOneLeft]{bondUone.east}{bondPhiOne.west}
\tnjoin[name=bondOneRight]{bondPhiOne.east}{bondUtwo.west}
\tnjoin[name=bondOuterRight]{bondUtwo.east}{bondRight.west}
\tnjoin[name=bondPhysicalJoinZero]
{bondUzero.north}{bondPhysicalZero.south}
\tnjoin[name=bondPhysicalJoinOne]
{bondUone.north}{bondPhysicalOne.south}
\tnjoin[name=bondPhysicalJoinTwo]
{bondUtwo.north}{bondPhysicalTwo.south}
\tnregion[slot=selected, label={$\supp(P_{01})$},
label pos=south,
name=bondProjectorZero]{bondPhiZero}
\tnregion[slot=selected, label={$\supp(P_{12})$},
label pos=south,
name=bondProjectorOne]{bondPhiOne}
\end{tenkzfree}](tenkz_svg/tenkz-8f771ece57c733f0.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 [ CPGSV16 , 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 [ CPGSV16 , 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 first identity 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 the two range identities. 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 [ CPGSV16 , Definition 3.9, Equations (3.16)–(3.18), and Appendix B ] . It does not assert commutation for every chain length.
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 14.1.42. 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{tenkzfree}
\tnput[boundary, ports={east:virtual}]
{transportLeft}{(-12mm,0)}{}
\tnput[box, ports={west:virtual,east:virtual,north:physical}]
{transportUzero}{(0,0)}{U_0}
\tnput[dot, ports={west:virtual,east:virtual},
label pos=south]
{transportPhiZero}{(16mm,0)}{\varphi_{01}}
\tnput[box, ports={west:virtual,east:virtual,north:physical}]
{transportUone}{(32mm,0)}{U_1}
\tnput[dot, ports={west:virtual,east:virtual},
label pos=south]
{transportPhiOne}{(48mm,0)}{\varphi_{12}}
\tnput[box, ports={west:virtual,east:virtual,north:physical}]
{transportUtwo}{(64mm,0)}{U_2}
\tnput[boundary, ports={west:virtual}]
{transportRight}{(76mm,0)}{}
\tnput[boundary, ports={south:physical}]
{transportPhysicalZero}{(0,14mm)}{}
\tnput[boundary, ports={south:physical}]
{transportPhysicalOne}{(32mm,14mm)}{}
\tnput[boundary, ports={south:physical}]
{transportPhysicalTwo}{(64mm,14mm)}{}
\tnjoin[name=transportOuterLeft]
{transportLeft.east}{transportUzero.west}
\tnjoin[name=transportZeroLeft]
{transportUzero.east}{transportPhiZero.west}
\tnjoin[name=transportZeroRight]
{transportPhiZero.east}{transportUone.west}
\tnjoin[name=transportOneLeft]
{transportUone.east}{transportPhiOne.west}
\tnjoin[name=transportOneRight]
{transportPhiOne.east}{transportUtwo.west}
\tnjoin[name=transportOuterRight]
{transportUtwo.east}{transportRight.west}
\tnjoin[name=transportPhysicalJoinZero]
{transportUzero.north}{transportPhysicalZero.south}
\tnjoin[name=transportPhysicalJoinOne]
{transportUone.north}{transportPhysicalOne.south}
\tnjoin[name=transportPhysicalJoinTwo]
{transportUtwo.north}{transportPhysicalTwo.south}
\tnregion[slot=selected, label={$\supp(P_{01})$},
label pos=south,
name=transportVirtualZero]{transportPhiZero}
\tnregion[slot=selected, label={$\supp(P_{12})$},
label pos=south,
name=transportVirtualOne]{transportPhiOne}
\tnregion[slot=secondary, outline,
label={$\supp((P_2)_{01})$},
label pos=north west, name=transportPhysicalRegionZero]
{transportUzero,transportUone,transportZeroLeft,
transportZeroRight,transportVirtualZero}
\tnregion[slot=secondary, outline,
label={$\supp((P_2)_{12})$},
label pos=north east, name=transportPhysicalRegionOne]
{transportUone,transportUtwo,transportOneLeft,
transportOneRight,transportVirtualOne}
\end{tenkzfree}](tenkz_svg/tenkz-fba65ba91d819147.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, before the \(XB\)-lift, the same coefficient-space representative, \(\widehat Q_{XB}=\widehat Q_{AX}\). The source projector \(Q_{XB}\) of Definition D.2 acts on \(\mathcal H_X\otimes \mathcal H_B\). The displayed operator is the coefficient representative later placed on the \(XB\) face.
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 [ CPGSV16 , 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 14.1.2 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 [ CPGSV16 , 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 14.1.2 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 14.1.1. This is the coefficient-space analog of the projector-and-commutator part of [ CPGSV16 , Definition D.2 ] . The hatted maps have not yet been identified with the source projectors on the explicitly factored spaces. By itself the result does not prove the kernel-intersection equation in that definition and makes no assertion for every chain length. The coefficient-space kernel equation is established in the next two results.
Both coefficient representatives are the complement \(1-P_2\) of the two-site basic support projector. Their idempotency was established in Lemma 14.1.55. 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 [ CPGSV16 , 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 [ CPGSV16 , 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.
For an Appendix B structural datum and a periodic chain of length \(N{\gt}2\), set \(p_i=h_i^{(N)}\) for \(i\in \mathbb Z/N\mathbb Z\), where \(h_i^{(N)}\) is the translated length-two parent interaction. These maps are idempotent and commute pairwise: adjacent commutation follows from Theorem 14.1.60, while disjoint length-two interactions commute by locality.
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 [ CPGSV16 , Definition 3.9 ] remain separate.
The Appendix B structural theorem supplies the datum used in Definition 14.1.61. Its commuting idempotents are precisely the translated parent interactions. The standard frustration-free identity shows that each interaction annihilates the periodic matrix product vector.
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 14.1.1.
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 14.1.1. 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 14.1.66.
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 14.1.65, 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
Thus the one-pair case of the factorization is already verified for the core tensor: \(V^{(2)}(\Lambda U)(\sigma )=\phi _2(\sigma _0,\sigma _1)\). The even-chain physical-pair factorization is the following condition, for a positive number \(n\) of pairs,
The case \(n=0\) is not part of this condition: the empty-chain MPV coefficient is the virtual bond dimension, whereas the empty product on the right-hand side is \(1\). The first equality, \(V^{(L)}(\Lambda U)_\sigma =C_L(\sigma )\), is the trace expansion of the closed matrix product and is the coefficient form of \(U^{\otimes L}\varphi _j^{\otimes L}\). The remaining coefficient step is to relate this source expression, with the physical word \(\sigma \) fixed, to the even-chain physical-pair factorization above. 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 14.1.56 constructs the adjacent physical support projectors from the basic-vector expression and proves the overlapping relation
Theorem 14.1.59 transports this identity around a periodic chain. Consequently, for every \(N{\gt}2\) and every pair of sites \(i,j\in \mathbb {Z}/N\mathbb {Z}\), \(h_i(A,2)h_j(A,2) = h_j(A,2)h_i(A,2)\). Definition 14.1.61 collects these maps into a commuting idempotent family; idempotency is Lemma 13.3.13. Together with Lemma 13.3.19, this gives the zero-energy equation for \(V^{(N)}(A)\). Neither conclusion requires the even-chain physical-pair factorization displayed above. That factorization remains a separate conditional coefficient statement. The full source implication additionally requires removal of the left-canonical hypothesis and the ground-space spanning equation. The coefficient-space analog of the local kernel-intersection equation is Theorem 14.1.58; identifying its hatted maps with the source projectors remains separate and is not needed for the chain commutator.
Suppose the conditional even-chain hypotheses give commuting two-site parent terms for a tensor \(B\). If the nearest-neighbor parent Hamiltonian also satisfies the ground-space spanning equation of Definition 14.1.15, with BNT components \(A_1,\ldots ,A_g\), then \(B\) satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition of Definition 14.1.17.
In particular, if a normal left-canonical RFP tensor has nonzero bond dimension, the same conditional Appendix B hypotheses, and the same ground-space spanning equation, then it satisfies the full all-chain condition in [ CPGSV16 , Theorem 3.10(iii) ] .
The conditional hypotheses give \(h_i h_j=h_j h_i\) for all translated two-site local terms. Lemma 13.3.19 gives \(h_iV^{(N)}(B)=0\). For every \(N{\gt}2\), the spanning hypothesis supplies \(\ker H_2^{(N)}(B) = \operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \). These are exactly the three clauses in Definition 14.1.16, uniformly in \(N\).
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 [ Bei12 , 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 first identity below holds on the first pair and the second on the last pair:
This is the spatial decomposition of [ Bei12 , 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 14.1.72.
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 [ Bei12 , 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 [ Bei12 , 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 14.1.74.
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 14.1.75 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 [ Bei12 , 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 [ Bei12 , 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.
For the convention that an \(L\)-site interaction is zero when \(L{\gt}N\), the two-site parent Hamiltonian on a one-site periodic chain satisfies
This convention is local to the present development. The two-site terms in [ Bei12 , Section III ] , source lines 451–514, are not assigned an action when both site labels denote the same site. The nearest-neighbor condition in [ CPGSV16 , Theorem 3.10(iii) ] , source lines 534–540, only concerns \(N{\gt}2\).
Since \(2{\gt}1\), the single translated interaction is zero by the stated convention. Its sum is therefore zero, 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 [ Bei12 , 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 14.1.70.
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. This is only a membership assertion; it does not assert that the loop states span the 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 14.1.90 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 14.1.91. Its dimension is the number of positive loops, which equals \(\dim \ker H_N\) by Corollary 14.1.81. 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 14.1.17: 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.
This is the normal left-canonical representative case of the reverse implication in Theorem 3.10 of [ CPGSV16 ] . The source theorem is stated for a tensor in canonical form; the unrestricted canonical-form statement is not asserted here.
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.
14.2 Decorrelation and the idempotent-product parent-commuting condition
This section develops the idempotent-product formulation used for the decorrelation implication from Appendix D. It is not the full source condition. In [ CPGSV16 , Appendix D.2 ] , the parent commuting Hamiltonian condition is stated for local projectors \(Q_{AX}\) and \(Q_{XB}\). Writing hats for their lifts to the tripartite space, the condition is
Writing \(P_{AX}=1-Q_{AX}\) and \(P_{XB}=1-Q_{XB}\) gives, for the idempotent \(P_K\) used below, the full-space identity \(\widehat P_{AX}\widehat P_{XB}=P_K\). The following definition records this idempotent-product consequence abstractly, not the tensor-product locality, orthogonal-projector, or ground-space-intersection hypotheses of Appendix D.2.
The following preparatory result is independent of the decorrelation condition. 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. These are the support-projector identities used in [ CPGSV16 , Appendix D.2, lines 2225–2235 ] .
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 every operator \(M\) on \(H_L\) and every operator \(\rho \) on \(H_L\otimes H_R\), one has \(\operatorname{tr}((M\otimes \mathbb {1}_R)\rho )=\operatorname{tr}(M\operatorname{tr}_R\rho )\).
Expand both traces in product bases and sum first over the identity entry on \(H_R\).
Let \(\rho \) be positive semidefinite on \(H_L\otimes H_R\), and let \(P_L\) be the orthogonal projector onto the support of \(\operatorname{tr}_R\rho \). Then \((P_L\otimes \mathbb {1}_R)\rho =\rho \).
Set \(Q_L=\mathbb {1}_L-P_L\). Since \(Q_L\operatorname{tr}_R\rho =0\), the adjoint identity gives
Positivity of \(\rho \) then implies \((Q_L\otimes \mathbb {1}_R)\rho =0\). Finally, \(\mathbb {1}_{L\otimes R}-(Q_L\otimes \mathbb {1}_R)=P_L\otimes \mathbb {1}_R\) gives the claim.
Under the same assumptions, \(\rho (P_L\otimes \mathbb {1}_R)=\rho \). For \(H_L=H_A\otimes H_X\) and \(H_R=H_B\), the two absorption identities are precisely the \(P_{AX}\) support-projector identities in [ CPGSV16 , Appendix D.2, lines 2228–2235 ] .
Apply \((\cdot )^\dagger \) to the left absorption identity. Since \(\rho ^\dagger =\rho \) and \((P_L\otimes \mathbb {1}_R)^\dagger =P_L\otimes \mathbb {1}_R\),
For an operator \(\rho \) on \(H_L\otimes H_R\), define \((\operatorname{tr}_L\rho )_{r,s}:=\sum _l\rho _{(l,r),(l,s)}\).
If \(\rho =\rho ^\dagger \), then \(\operatorname{tr}_L\rho =(\operatorname{tr}_L\rho )^\dagger \).
Entrywise Hermiticity gives
For finite-dimensional \(H_L\) and \(H_R\), if \(\rho \geq 0\), then \(\operatorname{tr}_L\rho \geq 0\).
If \(\rho ^{(k)}\) is the principal submatrix indexed by \(\{ k\} \times H_R\), then
Each \(\rho ^{(k)}\) is positive semidefinite, and hence so is their finite sum.
For every operator \(M\) on \(H_R\), one has \(\operatorname{tr}((\mathbb {1}_L\otimes M)\rho )=\operatorname{tr}(M\operatorname{tr}_L\rho )\).
Expanding in product bases gives
Let \(\rho \) be positive semidefinite. If \(P_R\) is the support projector of \(\operatorname{tr}_L\rho \), then \((\mathbb {1}_L\otimes P_R)\rho =\rho \).
Set \(Q=\mathbb {1}_R-P_R\). The preceding trace identity gives
since \(P_R\) is the support projector of \(\operatorname{tr}_L\rho \). Positivity of \(\rho \) then gives \((\mathbb {1}_L\otimes Q)\rho =0\), and hence \((\mathbb {1}_L\otimes P_R)\rho =\rho \).
Under the same assumptions, \(\rho (\mathbb {1}_L\otimes P_R)=\rho \).
Take adjoints in the left absorption identity.
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
The first and fourth formulas 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 [ CPGSV16 , 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 [ CPGSV16 , 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
Here the second equality uses the support-absorption identities and (??). The same calculation with the two factors reversed, using (??), gives (??).
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 [ CPGSV16 , 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 [ CPGSV16 , 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 (??); 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 14.2.15.
The structure used below records only the idempotent-product consequence of the parent commuting Hamiltonian condition of [ CPGSV16 , 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 14.2.41 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 (??) 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 [ CPGSV16 , Appendix D, Section D.2 ] ; it is not the full source parent commuting Hamiltonian condition.
Apply Lemma 14.2.43 to \(P_{AX}\) and \(P_{XB}\), and substitute \(P_{AX}\circ P_{XB}=P_K\) to obtain (??).
If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then
This is the algebraic consequence corresponding to equation (D.2) from [ CPGSV16 ] : \(K_{AXB}=(K_{AX}\otimes H_B)\cap (H_A\otimes K_{XB})\). 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 (??).
14.3 Spectral gap
A frustration-free Hamiltonian is gapped if the smallest nonzero eigenvalue, equivalently the gap above the ground space, is bounded away from zero uniformly in the chain length \(N\). For MPS parent Hamiltonians, the martingale method of Fannes–Nachtergaele–Werner and Nachtergaele [ FNW92 ; Nac96 ] , as recalled by Cirac–Perez-Garcia–Schuch–Verstraete [ CPGSV21 , Section IV.C ] and in the later formulation of Kastoryano and Lucia [ KL18 ] , reduces a uniform gap to an anticommutator estimate for overlapping local terms. Principal angles between the corresponding local ground spaces give the geometric interpretation of this estimate. The results below state the anticommutator condition for the cyclic-window local terms and also record cyclic norm-compression estimates as sufficient stronger hypotheses.
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 (??): \(h_{i,\mathrm{ES}}\) 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. The positivity result above 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 14.3.10 identifies the vanishing of this local projector with membership in \(G_L^{\mathrm{ES}}(A)\).
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 14.3.11 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 14.3.12. 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 14.3.11 translates the two local kernel assumptions into the left and right \(L\)-site ground conditions. The open-chain intersection property, Theorem 13.4.4.1, 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. The preceding projection-geometry identity for commuting symmetric projections gives the non-negative ordered cross term.
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
14.3.1 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 (??) is supplied.
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 14.3.1.4.
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 14.3.1.3.
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 (??). The finite-overlap row-sum reduction then applies. For the strict form, take \(\gamma =1-\eta m\).
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 (??) 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 14.3.1.1 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 14.3.1.1 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 \(P_i=h_{i,\mathrm{ES}}\).
For a fixed chain length \(N\), let \(m{\gt}0\) and \(\gamma \le 1\). Suppose an overlap predicate on local windows has at most \(m\) overlapping off-diagonal windows in each row, non-overlapping local terms have non-negative ordered cross terms, and overlapping local terms satisfy
Then \(H_{N,L}^{\mathrm{ES}}{}^2\ge \gamma H_{N,L}^{\mathrm{ES}}\) as a quadratic form. The same conclusion holds from a separate coefficient \(\eta \) satisfying \(\eta \le (1-\gamma )/m\).
Apply the finite-overlap norm-compression projection reduction to the family \(P_i=h_{i,\mathrm{ES}}\). If a separate coefficient \(\eta \) is used, first weaken the compression estimate using \(\eta \le (1-\gamma )/m\).
14.3.2 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 14.3.1.1 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 14.3.2.6 gives the row-cardinality bound \(m=2(L-1)\) for the overlap relation among length-\(L\) local windows, and Lemma 14.3.9 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 14.3.2.5 gives the required non-negative ordered cross term. Apply Lemma 14.3.2.7 with these facts and the overlapping-window estimate (??).
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 \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
Apply Lemma 14.3.2.8. Its non-overlap positivity and finite-family hypotheses are already internal to the preceding reduction; the overlapping estimate (??) supplies its remaining ordered cross-term condition.
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 \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
Fix \(L{\gt}1\). Suppose uniformly for every \(N\ge 2L\) and every overlapping off-diagonal pair that
Then the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).
For an overlapping pair, Cauchy–Schwarz and \(h_{i,\mathrm{ES}}^2=h_{i,\mathrm{ES}}\) give
This is the ordered lower bound used by the preceding overlapping-cyclic martingale reduction.
Fix \(L{\gt}1\), and let \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) and every overlapping off-diagonal pair that
If \(\gamma {\gt}0\), \(\gamma \le 1\), and \(\eta \le (1-\gamma )/m\), then \(0{\lt}\gamma \) and, for every admissible \(N\) and every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). In particular, if \(\eta \ge 0\) and \(\eta m{\lt}1\), then \(0{\lt}1-\eta m\) and, for every admissible \(N\) and every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \((1-\eta m)\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). This strict compression form is a norm-compression sufficient condition, not the principal-angle estimate itself. It is a sufficient strengthening of the anticommutator martingale estimate in Cirac–Perez-Garcia–Schuch–Verstraete [ CPGSV21 , Section IV.C ] . It does not prove the remaining MPS-specific estimate; it isolates the finite-row martingale argument once such a strengthening is supplied.
The overlap-cardinality lemma gives at most \(m=2(L-1)\) overlapping off-diagonal neighbours in each row, while the non-overlap positivity lemma handles disjoint supports. The finite-overlap norm-compression reduction gives the quadratic-form estimate with the chosen \(\gamma \), and the positive quadratic-form criterion converts it to the norm lower bound. The strict form is the same statement with \(\gamma =1-\eta m\).
Fix \(L{\gt}1\), and let \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) and every overlapping off-diagonal pair that
that is, a bound \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| \le \eta \) on the operator-product norm. If \(\gamma {\gt}0\), \(\gamma \le 1\), and \(\eta \le (1-\gamma )/m\), then \(0{\lt}\gamma \) and, for every admissible \(N\) and every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). In particular, if \(\eta \ge 0\) and \(\eta m{\lt}1\), then \(0{\lt}1-\eta m\) and the same norm bound holds with \(\gamma =1-\eta m\).
This is the symmetric (operator-product) counterpart of Lemma 14.3.2.12, obtained from the anticommutator conversion recorded in Remark 14.3.1.2. Like that lemma it is a conditional reduction: the excitation projections satisfy \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| =1\) whenever the overlapping ranges share a common nonzero vector, so the hypothesis is unsatisfiable for \(\eta {\lt}1\) in that case. It isolates the finite-row martingale argument once such a bound is supplied; the gap with the cited principal-angle estimates is documented in the paper-gap note docs/paper-gaps/cpgsv21_martingale_overlap.tex.
The overlap-cardinality lemma gives at most \(m=2(L-1)\) overlapping off-diagonal neighbours in each row, while the non-overlap positivity lemma handles disjoint supports. The symmetric anticommutator conversion of Remark 14.3.1.2 turns the operator-product bound into the per-pair anticommutator estimate with coefficient \((1-\gamma )/m\), using the hypothesis \(\eta \le (1-\gamma )/m\); the anticommutator finite-overlap reduction then gives the quadratic-form estimate with the chosen \(\gamma \), and the positive quadratic-form criterion converts it to the norm lower bound. The strict form is the same statement with \(\gamma =1-\eta m\), so the coefficient is then \(\eta \).
14.3.3 Martingale gap consequences
Consider a frustration-free Hamiltonian \(H=\sum _{i=1}^n h_i\) with projector interaction terms, \(h_i^2=h_i\). Suppose that for every unordered pair \(\{ i,j\} \) of overlapping terms there exist constants \(c_{\{ i,j\} } \ge 0\) and \(\gamma {\gt} 0\) satisfying
and suppose further that the coefficients attached to the terms overlapping \(h_i\) satisfy
for every \(i\). Then the smallest nonzero eigenvalue of \(H\) satisfies \(\lambda _{\min }^+(H) \ge \gamma \).
The relevant geometry is a family of translated length-\(L\) projector windows on the ring; two terms \(h_i\) and \(h_j\) interact only when their supports overlap, equivalently when the cyclic distance \(d_N(i,j)\) is smaller than \(L\). From \(h_i^2=h_i\),
Non-overlapping projectors commute, \(h_i h_j=h_j h_i\), so their product \(h_i h_j\) is itself a projector and hence PSD; thus the last sum is \(\ge 0\). For the overlapping sum, the martingale condition (??) gives \(h_i h_j+h_j h_i\ge -c_{\{ i,j\} }(1-\gamma )(h_i+h_j)\). Summing over unordered pairs \(\{ i,j\} \) gives
Collecting terms for each \(h_i\) gives
Here the last inequality uses the row-sum hypothesis \(\sum _{\substack {j\ne i\\ \{ i,j\} \text{ overlapping}}} c_{\{ i,j\} }\le 1\). Combining these estimates gives \(H^2\ge H-(1-\gamma )H=\gamma H\). For any eigenvector with \(H\psi =\lambda \psi \) and \(\lambda {\gt}0\), this gives \(\lambda ^2\ge \gamma \lambda \), hence \(\lambda \ge \gamma \) [ KL18 ] .
Let \(A\) be injective and fix an interaction range \(L \ge 2\) as in Definition 13.3.3. Let \(h_L(A):=\Pi _{G_L(A)^\perp }\) be the \(L\)-site parent interaction, and for each chain length and site \(i\) let \(h_i\) denote its translate as in Definition 13.3.11. For every pair of overlapping terms \(h_i\) and \(h_j\) with \(0{\lt}d_N(i,j){\lt}L\), where \(d_N(i,j):=\min (|i-j|,\, N-|i-j|)\) is the cyclic distance on the ring (i.e. whose \(L\)-site supports share at least one site under periodic boundary conditions), the source martingale argument asks for constants \(c_{\{ i,j\} }\) and \(\gamma {\gt}0\), independent of \(N\), for which the martingale condition of Theorem 14.3.3.1 holds. Cirac–Perez-Garcia–Schuch–Verstraete [ CPGSV21 , Section IV.C ] use the martingale estimate \(h_i h_j+h_j h_i \ge -c_{ij}(1-\gamma )(h_i+h_j)\) and interpret it as a lower bound on the smallest non-zero principal angle between the local ground spaces \(\ker h_i\) and \(\ker h_j\). In the Hilbert-space formulation above, the source condition is the corresponding quadratic-form inequality
with row-summable coefficients. Lemma 14.3.1.10 gives the implication from this anticommutator estimate to the spectral gap. A directed norm-compression estimate, independent of \(N\), such as
for every overlapping off-diagonal pair is a sufficient stronger hypothesis; Theorem 14.3.3.5 records that consequence. (When \(L=1\) the interaction terms have disjoint supports, so there are no overlapping pairs and the spectral gap follows directly without the martingale argument.)
The cited martingale paragraph compares the anticommutator estimate with principal angles between the local ground spaces. If \(q_i:=\mathbb {1}-h_i\) and \(q_j:=\mathbb {1}-h_j\) denote the corresponding ground-space projectors, the Kastoryano–Lucia estimate cited there concerns \(\delta (\ell )=\| q_iq_j-P\| \), where \(P\) projects onto \(\ker (h_i+h_j)=\ker h_i\cap \ker h_j\) and \(\ell \) is the overlap length. What remains is to derive the anticommutator estimate above, with its blocking convention and constants, for the cyclic-window local terms. The intersection property identifies \(\ker h_i\cap \ker h_j\) with the parent ground space on the union of the two windows, but it does not by itself supply this numerical inequality. The directed excitation-projection estimate
remains useful as a sufficient strengthening of the source condition.
Row-sum bound. Since \(L\ge 2\), each site \(i\) overlaps with at most \(2(L-1)\) neighbours (those \(j\) with \(0{\lt}d_N(i,j){\lt}L\)). Set \(c_{\{ i,j\} }:=\tfrac {1}{2(L-1)}\) for every unordered overlapping pair and \(c_{\{ i,j\} }:=0\) otherwise. These constants depend only on the cyclic distance \(d_N(i,j)\). Then
satisfying the row-sum hypothesis of Theorem 14.3.3.1.
Anticommutator constant. It remains to compare the cited principal-angle estimates with the cyclic-window anticommutator statement \(h_i h_j+h_jh_i \ge -c_{\{ i,j\} }(1-\gamma )(h_i+h_j)\). If this estimate is obtained with the displayed row-summable coefficients, Lemma 14.3.1.10 gives the gap. A sufficient stronger route is the norm-compression statement
If such an \(\eta \) is obtained, Theorem 14.3.3.5 gives the gap constant \(1-\eta \, 2(L-1)\). The frequently displayed fixed-coefficient compression form is the specialization with
In the corresponding anticommutator form the martingale condition is
for every overlapping off-diagonal pair, and a sufficient one-sided strengthening is
A qualitative statement that the relevant smallest non-zero principal angles are positive is not by itself enough: the martingale reduction needs a numerical anticommutator bound with row-summable coefficients, or a stronger norm-compression bound with \(\eta \, 2(L-1){\lt}1\). Thus the remaining comparison is whether the principal-angle estimates cited in [ FNW92 ; Nac96 ; KL18 ] supply such a cyclic-window estimate, with constants independent of \(N\), under the convention used here.
Let \(A\) be a tensor and let \(L{\gt}1\). Suppose that, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,
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 local averaging, positivity, and symmetric-projection statements are now established in Lemmas 14.3.7, 14.3.8, and 14.3.9. Lemma 14.3.1.1 reduces the present theorem to the remaining MPS-specific task: deriving the uniform quadratic-form estimate. Lemma 14.3.1.9 establishes the finite-sum algebra from ordered cross-term row bounds, while Lemma 14.3.1.10 records the source anticommutator martingale condition. Lemmas 14.3.1.12 and 14.3.2.7 state the common finite-overlap ordered-cross-term specialization with \(m=2(L-1)\). Lemma 14.3.17 supplies the non-overlap positivity statement for site-disjoint windows, and Lemma 14.3.2.5 translates this to the concrete support-disjoint cyclic windows. Lemma 14.3.2.6 gives the finite-overlap row-cardinality bound. Lemma 14.3.2.8 combines these results with the finite-overlap reduction, so the only additional input is the ordered anticommutator or cross-term estimate for overlapping cyclic windows. Lemma 14.3.2.9 states the same overlap-only formulation, and Lemma 14.3.2.11 states a sufficient norm-compression formulation. Equivalently, this is the constant-compression theorem below with \(\eta = \left(1-\frac{1}{4L}\right)\frac{1}{2(L-1)}\). Since \(1-\eta \, 2(L-1)=1/(4L)\), the displayed hypothesis gives the stated lower bound with \(\gamma _L=1/(4L)\).
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 \(\gamma _L\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).
This is Lemma 14.3.2.10 stated as the public conditional gap theorem, with the anticommutator inequality retained as an explicit hypothesis.
Let \(L{\gt}1\). Put \(m=2(L-1)\). Suppose that there is a constant \(\eta \ge 0\) such that \(\eta m{\lt}1\), and suppose that, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,
Then the parent Hamiltonian has the gap constant \(1-\eta m{\gt}0\). Concretely, if \(N\ge 2L\) and \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), then \((1-\eta m)\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).
Lemma 14.3.2.12, with \(\gamma =1-\eta m\), applies to the displayed hypothesis
It gives \(0{\lt}1-\eta m\) and the stated norm lower bound. The preceding fixed-coefficient theorem is the specialization \(\eta =(1-1/(4L))/(2(L-1))\).
Let \(L{\gt}1\). Put \(m=2(L-1)\). Suppose there is a constant \(\eta \ge 0\) with \(\eta m{\lt}1\) and that, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,
that is, a bound \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| \le \eta \) on the operator-product norm rather than the directional bound of Theorem 14.3.3.5. Then the parent Hamiltonian has the gap constant \(1-\eta m{\gt}0\). Concretely, if \(N\ge 2L\) and \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), then \((1-\eta m)\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). This is a conditional reduction. The local terms \(h_{i,\mathrm{ES}}=p_i\) are the excitation projections. If their overlapping ranges have a common nonzero vector, then \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| =1\), so the hypothesis is unsatisfiable for \(\eta {\lt}1\) in that case. In degenerate cases, such as a zero excitation projection, the operator-product norm can be smaller. In any case this is not the quantity controlled by the cited principal-angle estimates. The comparison is documented in docs/paper-gaps/cpgsv21_martingale_overlap.tex.
Lemma 14.3.2.13, with \(\gamma =1-\eta m\), applies to the displayed operator-product hypothesis and gives \(0{\lt}1-\eta m\) together with the stated norm lower bound.
For a tensor \(A\) and a range \(L{\gt}1\), assume either the cyclic-window anticommutator estimate of Theorem 14.3.3.4 or the stronger overlapping cyclic-window norm-compression estimate of Theorem 14.3.3.3. 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 14.3.18, 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.
For a tensor \(A\) and a range \(L{\gt}1\), suppose there is a constant \(\eta \ge 0\) such that \(\eta \, 2(L-1){\lt}1\) and, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,
Then there exists \(\gamma {\gt}0\) such that, for every \(N\ge 2L\) and every vector \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).
This follows from Theorem 14.3.3.5 by taking \(\gamma =1-\eta \, 2(L-1)\).
For a tensor \(A\) and a range \(L{\gt}1\), suppose there is a constant \(\eta \ge 0\) such that \(\eta \, 2(L-1){\lt}1\) and, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,
Then there exists \(\gamma {\gt}0\) such that, for every \(N\ge 2L\) and every vector \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). As in Theorem 14.3.3.6, this is a conditional reduction: the operator-product hypothesis is unsatisfiable for any overlapping pair whose excitation ranges have a common nonzero vector, since then \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| =1\). The statement does not assert this norm identity in degenerate cases.
This follows from Theorem 14.3.3.6 by taking \(\gamma =1-\eta \, 2(L-1)\).