11 Parent-Hamiltonian Foundations: Injective and Normal Ground Spaces
For a translation-invariant matrix product state with tensor \(A\), the parent Hamiltonian is the translation-invariant sum of local terms, each the orthogonal projector onto the complement of the length-\(L\) local space \(G_L(A)=\operatorname {im}(\Gamma _L)\), where \(\Gamma _L(X)(\sigma )=\operatorname{tr}(A^\sigma X)\) [ 228 , 17 ] . We identify the \(N\)-site space with functions \(\{ 0,\ldots ,d{-}1\} ^N\to \mathbb {C}\) on the computational basis.
The local ground spaces satisfy an intersection property that ensures uniqueness of the periodic ground state \(V^{(N)}(A)\) for injective and normal matrix product states. For injective tensors with positive bond dimension \(D\ge 1\), system size \(N\ge 2\), and interaction range \(1{\lt}L\le N\), the parent Hamiltonian is frustration-free with unique ground state \(V^{(N)}(A)\). For normal tensors with positive bond dimension \(D \ge 1\) and injectivity length \(L_0\), uniqueness holds whenever \(L_0 {\lt} L \le N\) with \(N \ge L_0 + 1\); word-span injectivity after blocking determines the boundary-closure identities that match open-interval restrictions to the periodic chain.
11.1 The \(N\)-site Hilbert space
For local dimension \(d\) and chain length \(N\), the \(N\)-site Hilbert space is \(\mathcal H_{N}^{(d)}:=\{ 0,\ldots ,d{-}1\} ^{N}\to \mathbb {C}\), identified with functions on the computational basis.
11.2 Local ground space \(G_L(A)\)
Fix an MPS tensor \(A=\{ A^i\} _{i=0}^{d-1}\) with \(A^i\in M_{D}(\mathbb {C})\) and a block length \(L\ge 1\). For a word \(w=(i_1,\ldots ,i_L)\), write \(A^w:=A^{i_1}\cdots A^{i_L}\). The boundary-condition parametrization is
Here \(A^\sigma \) abbreviates \(A^{\sigma (0)}\cdots A^{\sigma (L-1)}\). By cyclicity of the trace, this is the same as the convention of [ 228 ] , namely \(\operatorname{tr}(XA^\sigma )\) for the local space \(\mathcal G_L^A\).
The map \(\Gamma _L\) in (1) is a \(\mathbb {C}\)-linear map from \(M_{D}(\mathbb {C})\) to \((\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\). In tensor-network notation,
![\begin{tenkz}[rows={wire}, cols=5, boundary=periodic]
\tn[skin=ring]{X} &
\tn[label pos=135, ports={90:physical:$\sigma_1$}]{A} &
\tn[label pos=135, ports={90:physical}]{A} & \tn[skin=dots]{} &
\tn[label pos=135, ports={90:physical:$\sigma_L$}]{A}
\tnmark[form=bracket]{(1,2) .. (1,5)}{$L$}
\end{tenkz}](tenkz_svg/tenkz-850da2de312b2b1e.svg)
The chain denotes the word tensor \(A^\sigma \) with open physical indices \(\sigma _1,\ldots ,\sigma _L\), and the red capsule denotes the boundary matrix \(X\) inserted on the closing virtual bond before taking the trace. The brace records the chain length \(L\).
For every boundary matrix \(X\in M_{D}(\mathbb {C})\) and every length-\(L\) word \(\sigma \), one has \(\Gamma _L(X)(\sigma )=\operatorname{tr}(A^\sigma X)\).
11.2.1 The FNW boundary convention
Fannes, Nachtergaele, and Werner fix the product convention \(v(\mu _1,\ldots ,\mu _N)=v(\mu _1)\cdots v(\mu _N)\) in [ 194 , Equation (5.4) ] and define their boundary map in [ 194 , Equation (5.5) ] . The exact identification with the map in Definition 11.2.1 uses \(A_\mu =v(\mu )^\dagger \) and reverses the order of the physical sites.
For a length-\(N\) configuration \(\sigma \), define
where \(0\leq j{\lt}N\). This is an involutive permutation of \(\{ 0,\ldots ,d-1\} ^{N}\).
Precomposition with \(r_N\) defines a complex-linear isomorphism
This is a linear map of coefficient spaces and does not require an inner product.
For every coefficient function \(\psi \) and every configuration \(\sigma \),
This is the definition of precomposition by the permutation \(r_N\).
For \(v=\{ v(\mu )\} _{\mu =0}^{d-1}\), write \(v^\sigma =v(\sigma (0))\cdots v(\sigma (N-1))\), with the empty word equal to \(I_D\). Define the complex-linear map \(F_N^v\colon M_{D}(\mathbb {C})\to (\{ 0,\ldots ,d{-}1\} ^{N}\to \mathbb {C})\) by
This is the boundary map of [ 194 , Equation (5.5) ] , with the word order fixed by [ 194 , Equation (5.4) ] .
For every \(B\in M_{D}(\mathbb {C})\) and every length-\(N\) configuration \(\sigma \),
Evaluate the defining linear map at \(B\) and then at \(\sigma \).
Set \(A_\mu =v(\mu )^\dagger \). Then
For \(N{\gt}0\), the word on the right is \(A_{\sigma (N-1)}\cdots A_{\sigma (0)}\).
Conjugate transposition reverses the product:
Thus the site order in (7) comes from the adjointed word itself.
With the exact convention \(A_\mu =v(\mu )^\dagger \), one has
For every \(B\in M_{D}(\mathbb {C})\) and every configuration \(\sigma \),
The ordered list associated with \(r_N(\sigma )\) is the reverse of the list associated with \(\sigma \), as in (2). Together with the adjointed word in Theorem 11.2.1.6, this gives the physical-site reversal. Trace cyclicity is used only in the third equality, where it moves the boundary matrix \(B\) across the word.
These boundary identifications are algebraic relations between matrix spaces and coefficient functions, independent of any state-dependent boundary inner product.
The local ground space is the image \(G_L(A):=\operatorname{range}(\Gamma _L)\subseteq (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\).
For every \(L\), one has \(\dim G_L(A)\le D^2\).
Since \(G_L(A)\) is the range of a linear map from \(M_{D}(\mathbb {C})\), one has \(\dim G_L(A)\le \dim M_{D}(\mathbb {C})=D^2\).
The ambient space of \(L\)-site tensors satisfies \(\dim (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})=d^L\).
If \(d^L{\gt}D^2\), then \(G_L(A)\) is a proper subspace of the local Hilbert space, namely \(G_L(A)\neq (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\).
11.3 Parent interaction and chain Hamiltonian
The canonical parent interaction at range \(L\) is
The sources also allow any positive local interaction with this kernel; the orthogonal projector is the canonical representative used here.
Let \(\iota _L\colon (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\simeq \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\) be the canonical computational-basis identification. The local ground space under this \(\ell ^2\) identification is \(G_L^{\mathrm{ES}}(A):=\iota _L(G_L(A)) \subseteq \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\).
For \(v\in \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\), one has \(v\in G_L^{\mathrm{ES}}(A)\) if and only if \(\iota _L^{-1}v\in G_L(A)\).
This is the defining equation \(G_L^{\mathrm{ES}}(A)=\iota _L(G_L(A))\).
Let \(U\colon M_{D}(\mathbb {C})\to \ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\) be Frobenius vectorization in column–row coordinates. The Hilbert-space boundary map is
Thus \(\Gamma _L^{\mathrm{ES}}\) has a Euclidean coordinate-space domain, whereas the original map \(\Gamma _L\) has matrix domain \(M_{D}(\mathbb {C})\).
Let \(U\colon M_{D}(\mathbb {C})\to \ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\) be Frobenius vectorization, with the target indexed in column–row order, and let \(\iota _L\) be the computational-basis identification on the physical coefficient space. Then, for every \(X\in M_{D}(\mathbb {C})\), \(\Gamma _L^{\mathrm{ES}}(U(X))=\iota _L(\Gamma _L(X))\).
By definition, \(\Gamma _L^{\mathrm{ES}}\) is the composite of \(U^{-1}\), the original boundary map \(\Gamma _L\), and \(\iota _L\). Applying this composite to \(U(X)\) and cancelling \(U^{-1}U\) gives the claimed identity.
Let \(e_{(b,a)}\) denote the Frobenius coordinate vector representing the virtual matrix unit \(E_{ab}\). Then \(\Gamma _L^{\mathrm{ES}}(e_{(b,a)})=\iota _L(\Gamma _L(E_{ab}))\).
Frobenius vectorization records a matrix entry \((a,b)\) at the column–row coordinate \((b,a)\), so \(U(E_{ab})=e_{(b,a)}\). Substitute this identity into the boundary-map transport theorem.
The range of \(\Gamma _L^{\mathrm{ES}}\) is exactly \(G_L^{\mathrm{ES}}(A)\).
Every virtual vector is \(U(X)\) for a unique matrix \(X\), and the transport identity sends it to \(\iota _L(\Gamma _L(X))\). These are exactly the vectors in \(\iota _L(\operatorname{range}\Gamma _L)=G_L^{\mathrm{ES}}(A)\), proving both inclusions.
If \(A\) is \(L\)-block injective, then \(\Gamma _L^{\mathrm{ES}}\) is injective.
Write each virtual vector uniquely as \(U(X)\) by Frobenius vectorization. The transport identity reduces equality of their images under \(\Gamma _L^{\mathrm{ES}}\) to equality of the images of the corresponding matrices under \(\Gamma _L\). Block injectivity makes the latter map injective.
Suppose that \(L_0{\gt}0\), that \(A\) is \(L_0\)-block injective, and that \(L\ge L_0\). Then \(\Gamma _L^{\mathrm{ES}}\) is injective.
Positive-length propagation makes \(A\) block injective at length \(L\). Theorem 11.3.7 then gives injectivity of \(\Gamma _L^{\mathrm{ES}}\).
The boundary Gram operator is
The FNW convention starts from an isometry \(V\colon \mathbb {C}^D\to \mathbb {C}^d\otimes \mathbb {C}^D\) written as \(V\xi =\sum _\mu |\mu \rangle \otimes v(\mu )^\dagger \xi \) in [ 194 , Equation (5.2) ] . Equations (5.3b) and (5.3d) of that paper give \(\sum _\mu v(\mu )v(\mu )^\dagger =I_D\) and \(E(B)=\sum _\mu v(\mu )Bv(\mu )^\dagger \). Under the identification \(A^\mu =v(\mu )^\dagger \), this map uses the adjoint tensor letters. The asymptotic identity \(E^n(B)\to \operatorname{tr}(\rho B)I_D\) in [ 194 , Equation (5.1) ] is a separate assertion and is not used in the convention identities below.
Let \(A\) be an MPS tensor, with \(A^\mu =v(\mu )^\dagger \) in the notation of [ 194 , Equations (5.2)–(5.3) ] . Its FNW transfer map is
Write \(\mathcal{E}_A^\sharp \) for the adjoint of \(\mathcal{E}_A\) with respect to the bilinear trace pairing, characterized by
Then \(E_{\mathrm{FNW},A}=\mathcal{E}_A^\sharp \). Here \(\sharp \) denotes the adjoint for \((X,Y)\mapsto \operatorname{tr}(XY)\), so the theorem is phrased as a bilinear trace-adjoint identity. For the unweighted Frobenius inner product, the same map also agrees with the Hilbert-space adjoint of \(\mathcal{E}_A\). This Frobenius adjoint is generally different from the adjoint for the \(\rho \)-weighted inner product in [ 194 , Equation (5.6) ] .
Expanding both maps and using cyclicity of trace gives
The final sum is \(\operatorname{tr}(XE_{\mathrm{FNW},A}(Y))\).
If \(A\) is left canonical, then
Under \(A^\mu =v(\mu )^\dagger \), this is the normalization \(\sum _\mu v(\mu )v(\mu )^\dagger =I_D\) from [ 194 , Equation (5.3b) ] .
Left canonicity makes \(\mathcal{E}_A\) trace-preserving. Therefore, for every \(X\),
Nondegeneracy of the trace pairing gives \(E_{\mathrm{FNW},A}(I_D)=I_D\).
For all \(\rho ,X\in M_{D}(\mathbb {C})\),
This is the explicit trace-pairing form of the convention \(A^\mu =v(\mu )^\dagger \) for the map in [ 194 , Equations (5.2)–(5.3) ] .
Substitute the trace-pairing adjoint identity and use cyclicity of trace.
These identities establish the algebraic equivalence between the two transfer-map conventions.
Let \(\rho \in M_{D}(\mathbb {C})\) satisfy \(\operatorname{tr}(\rho )\neq 0\). The normalized rank-one map associated with \(\rho \) is
If \(\operatorname{tr}(\rho )=1\), this becomes \(E_\infty ^\rho (B)=\operatorname{tr}(\rho B)I_D\), which is the limiting map in [ 194 , Equation (5.1) ] .
Define the fixed-point projection
Then \(E_\infty ^\rho =P_\rho ^\sharp \), where \(\sharp \) denotes the adjoint for the bilinear trace pairing. For the trace-one Hermitian density matrix used by FNW, the same map also agrees with the Hilbert-space adjoint for the unweighted Frobenius inner product. This Frobenius adjoint is generally different from the adjoint for the \(\rho \)-weighted inner product of [ 194 , Equation (5.6) ] .
For all \(X,B\in M_{D}(\mathbb {C})\),
Nondegeneracy of the trace pairing proves the equality of maps.
For every \(B\in M_{D}(\mathbb {C})\),
Substitute (23) and use \(\operatorname{tr}(\rho I_D)=\operatorname{tr}(\rho )\). The nonzero trace cancels the normalizing denominator.
As an endomorphism of \(M_{D}(\mathbb {C})\), the limiting map satisfies
Apply Theorem 11.3.16 to every matrix \(B\).
If \(A\) is left canonical, then the FNW observable map in (17) satisfies
Apply Theorem 11.3.12 to obtain \(E_{\mathrm{FNW},A}(I_D)=\sum _\mu (A^\mu )^\dagger A^\mu =I_D\). Hence
Suppose that \(\rho \) is stationary for the Schrödinger transfer map, \(\mathcal{E}_A(\rho )=\rho \). Then
Suppose that \(A\) is left canonical and that \(\mathcal{E}_A(\rho )=\rho \). For every integer \(n\geq 1\),
The restriction \(n\geq 1\) is essential. At \(n=0\), the two sides would be \(I\) and \(I-E_\infty ^\rho \), respectively, and \(E_\infty ^\rho (I_D)=I_D\).
Write \(E=E_{\mathrm{FNW},A}\) and \(P=E_\infty ^\rho \). The identities of Theorem 11.3.17, Theorem 11.3.18, and Theorem 11.3.19 give \(EP=PE=P^2=P\), and hence \(E^mP=P\) for every \(m\geq 0\). The case \(n=1\) is immediate. If \((E-P)^n=E^n-P\), then
Let \(A\) be a primitive MPS tensor of nonzero bond dimension with fixed-point witness \(\rho \). Then, for every integer \(n\geq 1\),
Primitivity supplies \(\operatorname{tr}(\rho )\neq 0\), left canonicity, and the stationary equation \(\mathcal{E}_A(\rho )=\rho \). Apply Theorem 11.3.20.
These algebraic relations express the spectral projection and remainder decomposition of the transfer map before introducing the \(\rho \)-weighted inner product.
For a complex-linear map \(\mathcal L\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), let \(R(\mathcal L)\) be the matrix with entries
The operator \(\mathscr R(\mathcal L)\) on \(\ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\) is represented by the matrix \(R(\mathcal L)\) in the standard orthonormal basis. Thus the row–column pair \(((b,a),(e,c))\) is sent explicitly to the Choi row–column pair \(((c,e),(a,b))\).
For complex-linear maps \(\mathcal L,\mathcal M\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\),
The defining matrix entries of the Choi reshuffling are linear in the map.
Let \(\mathcal L\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be complex-linear and let \(\mathbf e_{(j,i)}\) be the standard unit vector at \((j,i)\). For all \(a,b,c,e\in \{ 0,\ldots ,D{-}1\} \),
This is the coordinate identity in (35); it is not a Hilbert–Schmidt pairing of \(E_{ab}\) with \(\mathcal L(E_{ce})\).
Evaluate the matrix of \(\mathscr R(\mathcal L)\) on the two standard basis vectors.
Let \(I\) be a finite index set and let \(B=(B_{ij})_{i,j\in I}\) be a complex matrix. Then the squared operator norm of \(B\) on \(\ell ^2(I)\) satisfies
Apply Cauchy–Schwarz to each row of \(Bx\), sum over the rows, and take the supremum over unit vectors \(x\in \ell ^2(I)\).
Let \(I\) be a finite index set, let \(A=(A_{ij})_{i,j\in I}\) be a complex matrix, and let \(j\in I\). Then
Apply \(A\) to the unit vector supported at \(j\). Its image is the \(j\)th column of \(A\), while the vector itself has Euclidean norm one.
Let \(I\) be a finite index set and let \(A=(A_{ij})_{i,j\in I}\) be a complex matrix. Then
Sum Theorem 11.3.26 over all columns.
Let \(I\) be a finite index set. For every \(i,j\in I\), the matrix unit \(E_{ij}\) satisfies
The matrix unit sends the unit vector at \(j\) to the unit vector at \(i\) and sends every vector to a scalar multiple of that unit vector.
For every complex-linear map \(\mathcal L\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), the sum of squared entry norms of \(R(\mathcal L)\) equals that of the rectangular Choi matrix:
The two matrices differ only by a reshuffling of indices; the equality is entrywise Frobenius, not operator norm.
The map \(((b,a),(e,c))\mapsto ((c,e),(a,b))\) is an involution (bijection) on the product index space. Reindexing the sum of squared norms by this involution yields the claimed equality.
Let \(I\) be a finite index set and let \(\mathcal L\colon \mathbb C^{I\times I}\to \mathbb C^{I\times I}\) be complex-linear. Then
Let \(I\) be a finite index set, equip \(\mathbb C^{I\times I}\) with its \(L^2\) operator norm, and let \(\mathcal L\colon \mathbb C^{I\times I}\to \mathbb C^{I\times I}\) be complex-linear. Denote the operator norm of \(\mathcal L\) for these matrix norms by \(\lVert \mathcal L\rVert _{\mathrm{op},L^2}\). Then
Let \(A\) be a complex Banach algebra, let \(a\in A\), and let \(\lambda \geq 0\). If \(\rho _{\mathrm{spec}}(a){\lt}\lambda \), then there is a constant \(C_\lambda {\gt}0\) such that
for every \(n\in \mathbb {N}\). This operator-theoretic estimate provides the prescribed-rate geometric decay used for the peripheral-spectrum remainder of the transfer map in [ 194 , Lemma 5.2 ] .
Gelfand’s formula gives \(N\in \mathbb {N}\) such that \(\lVert a^n\rVert \leq \lambda ^n\) for every \(n\geq N\). Since the spectral radius is nonnegative and strictly smaller than \(\lambda \), one has \(\lambda {\gt}0\). Set
The eventual estimate proves the claim for \(n\geq N\), while the corresponding summand in \(C_\lambda \) proves it for \(n{\lt}N\).
Let \(V\) be a complex Banach space, let \(T\colon V\to V\) be a bounded linear operator, and let \(\lambda \geq 0\). If \(\rho _{\mathrm{spec}}(T){\lt}\lambda \), then there is a constant \(C_\lambda {\gt}0\) such that
for every \(n\in \mathbb {N}\) and \(x\in V\).
Apply Theorem 11.3.32 in the Banach algebra of bounded endomorphisms of \(V\), then use \(\lVert T^n x\rVert \leq \lVert T^n\rVert \lVert x\rVert \).
Let \(A\) be a primitive MPS tensor of nonzero bond dimension with fixed-point witness \(\rho \), and set \(R_A^\rho =E_{\mathrm{FNW},A}-E_\infty ^\rho \). If \(\nu \) is an eigenvalue of \(R_A^\rho \), then
This is the norm-independent spectral step in [ 194 , Lemma 5.2 ] .
The remainder \(R_A^\rho \) is the bilinear trace adjoint of \(\mathcal{E}_A-P_\rho \). A linear map and its trace adjoint have the same characteristic polynomial, hence the same eigenvalues. Primitivity places every eigenvalue of \(\mathcal{E}_A-P_\rho \) strictly inside the unit disk.
Let \(\rho \in M_{D}(\mathbb {C})\). Define
This is the factor multiplying the weighted remainder norm in [ 194 , Equations (5.9)–(5.10) ] .
If \(\rho \in M_{D}(\mathbb {C})\) is positive definite and \(D{\gt}0\), then \(c_\rho {\gt}0\).
The inverse \(\rho ^{-1}\) is positive definite, so its trace has strictly positive real part.
Let \(\rho \in M_{D}(\mathbb {C})\) be positive definite and let \(\operatorname{tr}(\rho )\neq 0\). Equip \(M_{D}(\mathbb {C})\) with \(\langle X,Y\rangle _\rho =\operatorname{tr}(\rho X^\dagger Y)\). For a linear map \(F\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), write
The weighted remainder spectral radius is
where the remainder is viewed as a bounded operator on this weighted Hilbert space.
Let \(A\) be primitive with nonzero bond dimension and positive-definite fixed-point witness \(\rho \). Then
In finite dimension the spectral radius is determined by the eigenvalues. Apply Theorem 11.3.34 in the weighted norm.
Suppose that \(\rho \) is positive definite and \(\operatorname{tr}(\rho )=1\). For \(n\in \mathbb {N}\), define
This is the trace-one transfer-remainder quantity formed from the factor and weighted norm in [ 194 , Equations (5.9)–(5.10) ] .
The weighted operator norm is nonnegative, and hence so is \(a(n)\) for every \(n\in \mathbb {N}\).
A norm is nonnegative, and the trace factor \(c_\rho \) is strictly positive.
Let \(A\) be primitive with nonzero bond dimension and positive-definite trace-one fixed point \(\rho \). Let \(\lambda \geq 0\) satisfy \(r_\rho (A){\lt}\lambda \). There is a constant \(C_\lambda {\gt}0\) such that, for every integer \(n\geq 1\),
This proves the prescribed-rate transfer-remainder estimate in [ 194 , Lemma 5.2 ] . The condition \(\lambda {\lt}1\) is needed only when one subsequently deduces decay to zero.
Apply Theorem 11.3.32 to the weighted remainder \(R_A^\rho \). This gives \(C_0{\gt}0\) with \(\lVert (R_A^\rho )^n\rVert _{\mathrm{op},\rho } \leq C_0\lambda ^n\). When \(n\geq 1\), Theorem 11.3.21 identifies \((R_A^\rho )^n\) with \(E_{\mathrm{FNW},A}^n-E_\infty ^\rho \). Multiply by \(c_\rho {\gt}0\) and take \(C_\lambda =c_\rho C_0\).
The theorem does not identify \(C_\lambda \) with \(k^2\) or identify the FNW auxiliary dimension \(k\) with the chosen bond dimension \(D\).
11.3.1 The weighted FNW boundary estimate
Let \(\rho \in M_{D}(\mathbb {C})\) be positive definite with \(\operatorname{tr}(\rho )=1\). Equip \(M_{D}(\mathbb {C})\) with the inner product \(\langle B,C\rangle _\rho =\operatorname{tr}(\rho B^\dagger C)\) from [ 194 , Equation (5.6) ] , and let \(E_{ik}\) denote the standard matrix unit. Throughout this subsection, \(A^\mu =v(\mu )^\dagger \).
Precomposition with the reversal \(r_N\) defines a unitary operator
A permutation of the standard coordinates preserves the Euclidean inner product, and evaluation gives (56).
The boundary map of [ 194 , Equation (5.5) ] defines a bounded linear operator
In the ground-space convention,
The algebraic identity \(F_N^v=R_N\circ \Gamma _N^A\) from Theorem 11.2.1.7 is unchanged after equipping the domain with the weighted norm and the codomain with the Euclidean norm. The reversal becomes the unitary operator \(\mathcal R_N\).
For all \(B,C\in M_{D}(\mathbb {C})\),
These are the two forms of the scalar-product identity in [ 194 , Equation (5.8) ] . Identity (61) preserves the source orientation \(B^\dagger E_{ik}C\).
For all \(B,C\in M_{D}(\mathbb {C})\),
The trace-one limiting map satisfies
Subtract this identity from (61).
For all \(B,C\in M_{D}(\mathbb {C})\),
This is the estimate in [ 194 , Equation (5.9) ] . It contains no factor depending only on the dimension.
Regard the sum in (62) as the trace of the endomorphism obtained by composing the transfer remainder with \(X\mapsto B^\dagger XC\). Weighted trace duality bounds this trace by \(\operatorname {Re}\operatorname{tr}(\rho ^{-1})\) times the weighted operator norm of the remainder and the two weighted boundary norms.
For every \(B\in M_{D}(\mathbb {C})\),
This recursion follows from the boundary formula in [ 194 , Equation (5.5) ] and is used in the proof of [ 194 , Lemma 5.3 ] . Here \(A^\mu =v(\mu )^\dagger \).
Split a configuration of length \(N+1\) into its first \(N\) entries and its final entry \(\mu \). The adjointed word identity gives \(F_{N+1}(B)(\sigma ,\mu )=F_N(BA^\mu )(\sigma )\). Summing the squared absolute values over \(\sigma \) and \(\mu \) proves (65).
Theorem 11.3.1.5 and Theorem 11.3.1.6 establish (64) and the right-boundary recursion (65) for [ 194 , Lemma 5.3 ] . They do not prove Lemma 6.2. In its source proof, Equation (6.5) contributes \(a(m)/a_-(m)\), while the subsequent orthogonality estimates contribute \(a(m)^2/a_-(m)\). Their sum is \(a(m)(1+a(m))/a_-(m)\).
11.3.2 The FNW lower boundary constant
Let \(D{\gt}0\), let \(\rho \in M_{D}(\mathbb {C})\) be positive definite with \(\operatorname{tr}(\rho )=1\), and equip \(M_{D}(\mathbb {C})\) with
Write \(F_N\) for the length-\(N\) FNW boundary map and let \(a(N)\) be the source mixing quantity in (53). This is the normalization used in [ 194 , Equations (5.9)–(5.10) ] .
For every \(B,C\in M_{D}(\mathbb {C})\),
In particular,
No sign assumption on \(a(N)\) is needed for the last inequality.
For \(B\neq 0\), define the boundary norm-square ratio
The lower boundary constant of [ 194 , Lemma 5.3 ] is
The weighted Gram operator on the virtual matrix Hilbert space is
The operator \(G_N\) is self-adjoint and positive. For every nonzero \(B\in M_{D}(\mathbb {C})\),
The adjoint identity gives \(\langle G_NB,B\rangle _\rho =\langle F_NB,F_NB\rangle \). Divide by \(\lVert B\rVert _\rho ^2\) and take the infimum over \(B\neq 0\). Self-adjointness and positivity follow directly from \(G_N=F_N^\dagger F_N\).
The real number \(a_-(N)\) is a spectral value of \(G_N\), and every \(z\in \operatorname {spec}(G_N)\) satisfies
Thus \(a_-(N)\) is the least spectral value in the sense of real parts. The spectrum is a subset of \(\mathbb {C}\), where no compatible order is fixed, so the assertion is stated as spectral membership together with (76).
The minimum of the Rayleigh quotient of a finite-dimensional self-adjoint operator is an eigenvalue. Conversely, if \(G_NB=zB\) with \(B\neq 0\), then self-adjointness makes \(z\) real and
Let \(m=\dim _{\mathbb {C}}M_{D}(\mathbb {C})\). There is an index \(i\in \{ 0,\ldots ,m-1\} \) such that
and \(a_-(N)\leq \sigma _j(F_N)^2\) for every \(j\in \{ 0,\ldots ,m-1\} \). The indices range only over the finite virtual domain dimension. No zero-padded sequence indexed by all natural numbers is used.
The squared singular values of \(F_N\) are the eigenvalues of \(F_N^\dagger F_N=G_N\), indexed by an orthonormal eigenbasis of the finite-dimensional domain. Apply Theorem 11.3.2.5.
For every \(B\in M_{D}(\mathbb {C})\),
More generally, a real number \(c\) satisfies
The lower boundary constant satisfies
Consequently, \(a_-(N){\gt}0\) whenever \(a(N){\lt}1\). This is the lower estimate in [ 194 , Lemma 5.3 ] . No equality \(a_-(N)=1-a(N)\) is asserted.
If \(\mathcal{E}_A(\rho )=\rho \), then every \(B\in M_{D}(\mathbb {C})\) satisfies
Expand the left-hand side using (67), cycle the trace, and sum over \(\mu \). The result is \(\operatorname {Re}\operatorname{tr}(\mathcal{E}_A(\rho )B^\dagger B)\), which equals the right-hand side by stationarity.
If \(\mathcal{E}_A(\rho )=\rho \), then
for every \(N\in \mathbb {N}\). Hence \(N\mapsto a_-(N)\) is nondecreasing. The same conclusion holds for every primitive MPS tensor with positive-definite fixed point \(\rho \).
Suppose that \(A\) is block injective at length \(L_0\). Then
If \(\mathcal{E}_A(\rho )=\rho \), then for every \(m\geq L_0\),
This is the positivity input used in the proof of [ 194 , Lemma 6.2 ] .
Injectivity makes every singular value of \(F_{L_0}\) positive. The least squared singular value is \(a_-(L_0)\) by Theorem 11.3.2.6. Monotonicity then gives \(a_-(m)\geq a_-(L_0){\gt}0\) for \(m\geq L_0\).
These results establish the lower-boundary assertions of [ 194 , Lemma 5.3 ] .
11.3.3 The linear overlap estimate in FNW Lemma 6.2
Fix consecutive block lengths \(\ell ,m,r\). Write \(\mu ^\ell \), \(\mu ^m\), and \(\mu ^r\) for configurations on these blocks. The physical configuration is \(\mu ^\ell \mu ^m\mu ^r\), with the first two blocks grouped before the right block. All virtual matrix families below use the \(\rho \)-weighted norm from (67).
For a finite configuration space \(S\), let
The canonical three-block bijection sends \(((\mu ^\ell ,\mu ^m),\mu ^r)\) to the concatenated configuration \(\mu ^\ell \mu ^m\mu ^r\) and has the evident inverse on such concatenations.
For \(\Phi \in \mathscr H_\rho (\{ 0,\ldots ,d-1\} ^r)\) and \(\Psi \in \mathscr H_\rho (\{ 0,\ldots ,d-1\} ^\ell )\), define physical vectors \(L_{\ell ,m,r}\Phi \) and \(R_{\ell ,m,r}\Psi \) by
For \(B\in M_{D}(\mathbb {C})\), define the special spectator families
These are the families obtained by splitting the length-\(\ell +m+r\) boundary vector at the two sides of the middle block.
Let \(v(\mu )\) denote the product of the FNW letters along a word \(\mu \). For each pair \((\mu ^\ell ,\mu ^r)\), set
More generally, for words \(u\) and \(\tau \) of lengths \(K\) and \(L\),
Thus the left word moves to the left of the boundary matrix and the right word moves to its right.
The weighted matrix identity
gives the source aggregates
Then
Consequently, the difference of these two pairings is the sum of the length-\(m\) boundary defects over all spectator pairs.
Reindex the physical inner product by the three consecutive blocks and apply the two boundary-word splitting identities. For the aggregate pairing, use \(\rho \rho ^{-1}=\rho ^{-1}\rho =I_D\) and cyclicity of trace, then distribute the two finite sums.
Suppose that \(A\) is left canonical and \(\mathcal{E}_A(\rho )=\rho \). For every \(N\geq 0\) and \(X\in M_{D}(\mathbb {C})\),
Consequently,
Unitality of \(E_{\mathrm{FNW},A}^N\) gives the first word sum. The trace-pairing identity and stationarity of \(\rho \) give the second. Summing over the remaining spectator index yields the two displayed identities.
Suppose that \(A\) is left canonical and \(\mathcal{E}_A(\rho )=\rho \). Then
Expand the finite words using the boundary convention from [ 194 , Equations (5.5) and (5.8) ] . Trace cyclicity gives the right overlap identity. Stationarity of \(\rho \) gives the left aggregate identity, while unitality of the FNW word map gives the right one and the two norm identities.
If \(\operatorname{tr}(\rho )=1\), then
If, in addition, \(A\) is left canonical and \(\mathcal{E}_A(\rho )=\rho \), then
The overlap norms satisfy
The optimal lower-boundary inequality also gives
If \(\mathcal{E}_A(\rho )=\rho \) and \(a_-(m){\gt}0\), monotonicity from \(m\) to \(\ell +m\) and from \(m\) to \(m+r\) gives
Reindex each physical norm by the spectator configuration. Apply (79) to every summand. Monotonicity gives \(a_-(m)\leq a_-(\ell +m)\) and \(a_-(m)\leq a_-(m+r)\). Multiply the resulting square-root estimates.
Suppose that \(\operatorname{tr}(\rho )=1\), \(A\) is left canonical, \(\mathcal{E}_A(\rho )=\rho \), and \(a_-(m){\gt}0\). Then
This is the linear estimate in [ 194 , Equation (6.5) ] . Positivity of the actual lower boundary constant is assumed directly.
11.3.4 The quadratic aggregate contribution in FNW Lemma 6.2
At length \(N\), let
This is the full open-chain ground-space range used in the orthogonality step of [ 194 , Lemma 6.2 ] .
Suppose that \(\operatorname{tr}(\rho )=1\), \(A\) is left canonical, \(\mathcal{E}_A(\rho )=\rho \), and both \(L_{\ell ,m,r}\Phi \) and \(R_{\ell ,m,r}\Psi \) are orthogonal to \(\mathcal G_{\ell +m+r}\). Then, for every \(B\in M_{D}(\mathbb {C})\),
Consequently,
In the estimate preceding the lower-boundary conversion in [ 194 , Equation (6.5) ] , test each overlap vector against the matching full-ground spectator family from Theorem 11.3.3.7. Orthogonality gives the left identity
In the right case, it gives
Applying equation (5.9) with (122) and (123) leaves the corresponding aggregate pairings with the stated bounds. Test once more against each aggregate itself to obtain the two norm estimates.
Under the hypotheses of Theorem 11.3.4.2, if \(a_-(m){\gt}0\), then
Under the hypotheses of Theorem 11.3.4.2, if \(a_-(m){\gt}0\), then
This is the quadratic contribution in the proof of [ 194 , Lemma 6.2 ] .
Apply Cauchy–Schwarz to the aggregate pairing and multiply the two one-sided bounds. Since \(\sqrt{a_-(m)}^2=a_-(m)\), the coefficient is exactly \(a(m)^2/a_-(m)\).
Under the hypotheses of Theorems 11.3.3.10 and 11.3.4.4,
The two source contributions are deliberately left unfactored.
Insert the aggregate pairing and apply the triangle inequality. The difference term is bounded by Equation (6.5), and the aggregate term is bounded by Theorem 11.3.4.4.
Write \(P^{\mathrm{tail}}_{r;m,\ell }\) and \(P^{\mathrm{left}}_{r+m;\ell }\) for the orthogonal projectors onto the reassociated tail-boundary range and the left-boundary range, respectively, and write \(G_{r+m+\ell }\) for the orthogonal projector onto the full ground space.
Let \(A\) have nonzero bond dimension, let \(\rho \) be positive definite with \(\operatorname{tr}(\rho )=1\), and suppose that \(A\) is left canonical and \(\mathcal E_A(\rho )=\rho \). Suppose that \(L_0\geq 1\) is an injective length. For every \(r\geq 0\), every \(\ell \geq 1\) and every \(m\geq L_0\),
In the source coordinates, let \(U\) and \(V\) be the two overlap ranges and let \(W\) be the full boundary range. Then \(W\subseteq U\cap V\). For \(x\in U\cap W^\perp \) and \(y\in V\cap W^\perp \), Theorem 11.3.4.5 gives
Projecting a vector first onto \(V\cap W^\perp \) and then onto \(U\cap W^\perp \) converts this relative-overlap bound into the same operator-norm bound for \(P_UP_V-P_W\). Global reversal is unitary and sends \(U\), \(V\), and \(W\) to the three ranges in the statement, so it preserves the norm.
First apply Theorem 11.3.4.6, which retains the two source contributions. Then use
Let \(A\) be primitive with positive-definite fixed point \(\rho \) and \(\operatorname{tr}(\rho )=1\). Then there is an injective length \(L_0\geq 1\) such that the bound (128) holds whenever \(m\geq L_0\) and \(\ell \geq 1\), for every \(r\geq 0\).
Choose a positive injective length from primitivity. Left canonicality and the fixed-point equation are part of the primitive hypotheses, so Theorem 11.3.4.6 applies.
11.4 The geometric form of the FNW projector estimate
Nachtergaele imports the projector estimate of Theorem 11.3.4.8 in the rational geometric form
stated in [ 225 , lines 1180–1194 ] and again for the best overlap constant \(A^\alpha _m\) in [ 225 , Equation (6.1) ] . This bound follows from the FNW coefficient \(a(m)(1+a(m))/a_-(m)\), the lower-boundary inequality \(1-a(m)\leq a_-(m)\), and the prescribed-rate estimate \(a(m)\leq c\lambda ^m\).
The source’s rate condition is now established by Theorem 12.8.46: every positive rate exceeding the moduli of the nonunit observable-transfer eigenvalues also exceeds the weighted spectral radius of the transfer remainder. In particular, the geometric estimate is available for every source-admissible rate \(0{\lt}\lambda {\lt}1\), with a positive rate-dependent prefactor.
The source’s simultaneous assertions that \(c=k^2\), the square of the auxiliary dimension, and that every admissible rate may be prescribed require a correction at the fixed intersection threshold. Theorem 12.8.1.16 gives a physical counterexample. The valid replacement retains the rate-dependent prefactor, or permits any prescribed positive prefactor after an additional onset depending on the state, rate, and prefactor, as proved in Theorem 12.8.48. See [ 135 ] .
Under the hypotheses of Theorem 11.3.4.6, suppose in addition that \(a(m)\leq c\lambda ^m{\lt}1\). Then
The FNW coefficient is \(a(m)(1+a(m))/a_-(m)\). Equation (5.9) gives \(1-a(m)\leq a_-(m)\), and \(a(m)\leq c\lambda ^m{\lt}1\) makes the left-hand side strictly positive, so the denominator may be replaced by \(1-c\lambda ^m\). The numerator is nondecreasing in \(a(m)\) on the nonnegative reals, so it is at most \(c\lambda ^m(1+c\lambda ^m)\).
Let \(A\) be primitive with positive-definite fixed point \(\rho \) and nonzero bond dimension. There are constants \(c{\gt}0\) and \(0{\lt}\lambda {\lt}1\) and an injective length \(L_0\geq 1\) such that, whenever \(m\geq L_0\) and \(c\lambda ^m{\lt}1\), the bound (133) holds for every prefix length \(r\) and every \(\ell \geq 1\).
Rescale the fixed point to unit trace; this changes neither the tensor nor the primitivity certificate. The weighted remainder has spectral radius below one, so there is a rate \(\lambda \) strictly between that radius and one; the prescribed-rate estimate supplies \(c{\gt}0\) with \(a(m)\leq c\lambda ^m\) for every \(m\geq 1\). Normality supplies a positive injective length. Apply the fixed-length geometric estimate.
A one-site suffix increment needs no reassociation, so the open-chain projector defect at overlap length \(l\) and prefix length \(K\) equals the whole-increment defect with \(r=K\), \(m=l\), and \(\ell =1\). Consequently, for a primitive tensor with positive-definite fixed point there are \(c{\gt}0\), \(0{\lt}\lambda {\lt}1\), and an injective length \(L_0\geq 1\) such that, whenever \(l\geq L_0\) and \(c\lambda ^l{\lt}1\), for every prefix length \(K\),
This establishes the projector defect bound (132) for primitive matrix product states.
The two projectors of the whole-increment display at \(\ell =1\) are the two projectors of the open-chain display. Indeed
where \(\operatorname{ran}\widetilde\Gamma ^{\mathrm{tail}}_{K;l,1}=\operatorname{ran}\Gamma ^{\mathrm{tail}}_{K,l+1}\) follows by (212) and the remaining subspace identifications hold by Theorem 12.3.5.17. Taking orthogonal projectors onto the three subspaces gives
so the two defect operators are equal. Substituting \(\ell =1\) in the geometric estimate gives the display.
Let \(V\) be a complex Banach space and let \(T\colon V\to V\) be a bounded linear operator. If \(\rho _{\mathrm{spec}}(T){\lt}1\), then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that
for every \(n\in \mathbb {N}\).
Choose \(r\) strictly between the spectral radius and one, then apply Theorem 11.3.32 with \(\lambda =r\).
Let \(V\) be a complex Banach space and let \(T\colon V\to V\) be a bounded linear operator. If \(\rho _{\mathrm{spec}}(T){\lt}1\), then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that
for every \(n\in \mathbb {N}\) and \(x\in V\).
Apply Theorem 11.4.4 and the defining operator-norm inequality to \(T^n x\).
Let \(\mathbf e_{(j,i)}:=\operatorname {single}_{(j,i)}(1)\) denote the standard unit vector in \(\ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\), and let \(\mathcal E_A\) be the transfer map. Then
Under Frobenius vectorization, \(\mathbf e_{(b,a)}=U(E_{ab})\) and \(\mathbf e_{(e,c)}=U(E_{ce})\), explaining the reversed coordinate order. Thus the Gram operator is the indicated Choi reshuffling; in particular it is not obtained by replacing the right-hand side with \(\langle E_{ab},\mathcal E_A^L(E_{ce})\rangle _{\mathrm{HS}}\).
Move the adjoint in \(\mathcal K_L=(\Gamma _L^{\mathrm{ES}})^\dagger \Gamma _L^{\mathrm{ES}}\) across the inner product and expand the two boundary-map coefficients. For each physical word \(\sigma \), the resulting summand is \((A^\sigma )_{ec}\, \overline{(A^\sigma )_{ba}}\). On the other hand, expanding the transfer power on \(E_{ca}\) gives
The defining Choi indexing \(J(\mathcal L)_{(c,e),(a,b)}=(\mathcal L(E_{ca}))_{eb}\) now yields the stated reshuffling.
The ground-space Gram operator is exactly the operator-level reshuffling of the Choi matrix of the iterated transfer map:
The equality is an equality of operators on \(\ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\). Its entries are those in (37), rather than Hilbert–Schmidt pairings with \(\mathcal E_A^L\).
The two operators have equal matrix coefficients on every pair of standard basis vectors. Hence they are equal.
Let \(A\) be an MPS tensor of bond dimension \(D\), let \(\rho \in M_{D}(\mathbb {C})\) satisfy \(\operatorname{tr}(\rho )\neq 0\), and let \(P_\rho \) be the fixed-point projection
Suppose that \(\mathcal E_A\) is trace-preserving, \(\mathcal E_A(\rho )=\rho \), and
Then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that, for every \(n\geq 1\),
No positivity, injectivity, or uniqueness hypothesis is imposed.
Write \(N=\mathcal E_A-P_\rho \). The power decomposition gives \(\mathcal E_A^n=P_\rho +N^n\) for \(n\geq 1\), while Theorem 11.4.4 gives \(\lVert N^n\rVert \leq Cr^n\). Use the exact Gram reshuffling identity and Theorem 11.3.23 to rewrite the left-hand side as \(\lVert \mathscr R(N^n)\rVert ^2\), then apply Theorem 11.3.31.
Let \(A\) be a primitive MPS tensor with fixed-point witness \(\rho \). Then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that, for every \(n\geq 1\),
Positive semidefiniteness and nonvanishing of \(\rho \) imply \(\operatorname{tr}(\rho )\neq 0\). The trace-preserving identity, fixed-point equation, and complementary spectral-radius bound are the remaining fields of the primitive-MPS hypothesis, so Theorem 11.4.8 applies.
Let \(A\) be an MPS tensor of bond dimension \(D\), let \(\rho \in M_{D}(\mathbb {C})\) satisfy \(\operatorname{tr}(\rho )\neq 0\), and let \(P_\rho \) be the corresponding fixed-point projection. If \(\mathcal E_A\) is trace-preserving, \(\mathcal E_A(\rho )=\rho \), and \(\rho _{\mathrm{spec}}(\mathcal E_A-P_\rho ){\lt}1\), then
in operator norm as \(n\to \infty \).
The right-hand side of (144) tends to zero because \(0{\lt}r{\lt}1\). Taking square roots gives the asserted norm convergence.
If \(A\) is a primitive MPS tensor with fixed-point witness \(\rho \), then
in operator norm as \(n\to \infty \).
The primitive-MPS data supply the nonzero trace, trace preservation, fixed-point equation, and complementary spectral-radius hypothesis of Theorem 11.4.10.
The limit in Theorem 11.4.10 is the reshuffled fixed-point projection \(\mathscr R(P_\rho )\), so the limiting metric is generally nonidentity. Under a positive-definite fixed-point hypothesis, the inverse estimates below perturb around \(\mathscr R(P_\rho )\) rather than around the identity.
Let \(\rho \in M_{D}(\mathbb {C})\) have nonzero trace and define \(P_\rho (X):=(\operatorname{tr}X/\operatorname{tr}\rho )\rho \). Then
If \(D{\gt}0\) and \(\rho \in M_{D}(\mathbb {C})\) is positive definite, then \(R(P_\rho )\) is positive definite and \(\mathscr R(P_\rho )\) is invertible, hence injective.
Positive definiteness gives \(\operatorname{tr}\rho {\gt}0\). By (148),
Hence its associated operator \(\mathscr R(P_\rho )\) is invertible.
Let \(K_0\) be invertible in a normed ring with convergent geometric series and \(\lVert I\rVert =1\). If \(a{\lt}1\) and
then \(K\) is invertible and
The same conclusion follows from \(\lVert K_0^{-1}\rVert \, \lVert K-K_0\rVert \le a\).
Put \(E:=K_0^{-1}(K-K_0)\). Then \(K=K_0(I+E)\) and \(\lVert E\rVert {\lt}1\), so
Submultiplicativity reduces the product-form hypothesis to (152).
Let \(K_0\) be invertible in a normed ring with convergent geometric series and \(\lVert I\rVert =1\). If \(a{\lt}1\) and (152) holds, then
The same estimate follows from \(\lVert K_0^{-1}\rVert \, \lVert K-K_0\rVert \le a\).
With \(E:=K_0^{-1}(K-K_0)\), the resolvent identity is
Since \(\lVert (I+E)^{-1}\rVert \le (1-a)^{-1}\) and \(\lVert E\rVert \le a\), taking norms gives (156). Submultiplicativity again gives the product-form variant.
Let \(K_i\) be a net in a normed ring with convergent geometric series. If \(K_i\to K_0\) and \(K_0\) is invertible, then
Write \(K_0\) as the image of a unit. Continuity of inversion on the unit group, followed by the convergence \(K_i\to K_0\), gives the result.
For an injective continuous linear map \(T\) with finite-dimensional domain, let \(S_T\) be the continuous endomorphism induced by the inverse linear equivalence of the Gram endomorphism \(T^\dagger T\).
Let \(\mathbb K\) be a real or complex scalar field, let \(E\) be a finite-dimensional complete \(\mathbb K\)-Hilbert space, and let \(F\) be a complete \(\mathbb K\)-Hilbert space. If \(T\colon E\to F\) is an injective continuous linear map, then the inverse Gram operator is a two-sided inverse of the Gram endomorphism:
Consequently, \(S_T\) is both the canonical inverse and the ring inverse of \(T^\dagger T\). The two pseudoinverse factors satisfy
By definition, \(S_T\) is induced by the inverse linear equivalence of the injective endomorphism \(T^\dagger T\). The inverse-equivalence identities give (159) and (160). Together these equations characterize the canonical inverse of a continuous linear endomorphism, which agrees with its inverse in the endomorphism ring. For the norm identities, use the \(C^*\)-identity and the two inverse equations to compute the squares of \(\lVert TS_T\rVert \) and \(\lVert S_TT^\dagger \rVert \).
For a positive-definite fixed point \(\rho \), Theorem 11.4.19 provides quantitative inverse bounds for the limiting metric \(\mathscr R(P_\rho )\).
Let \(A\) be a primitive MPS tensor of nonzero bond dimension whose fixed-point witness \(\rho \) is positive definite. Put
There are constants \(g,c,r{\gt}0\) with \(r{\lt}1\) and \(c=\lVert \mathcal I_\infty \rVert g\) such that, for every \(n\geq 1\),
Whenever \(cr^n{\lt}1\), the operator \(\mathcal K_n\) is invertible and
Start from the squared estimate \(\lVert \mathcal K_n-\mathcal K_\infty \rVert ^2 \leq D^3(Cr^n)^2\). Taking square roots gives \(g=\sqrt{D^3}\, C\) without any additional dimension loss, and hence (164). Positive definiteness makes \(\mathcal K_\infty \) invertible. The relative error is at most \(\lVert \mathcal I_\infty \rVert gr^n=cr^n\), so the two arbitrary-base Neumann estimates give (165) and (166).
Under the hypotheses of Theorem 11.4.19, there are constants \(g,c,r{\gt}0\) with \(r{\lt}1\) and \(c=\lVert \mathcal I_\infty \rVert g\) such that (164) holds for every \(n\geq 1\). If \(cr^n{\lt}1\), then the physical boundary map \(\Gamma _n^{\mathrm{ES}}\) is injective, its inverse Gram operator equals \(\mathcal K_n^{-1}\), and
Invertibility of \((\Gamma _n^{\mathrm{ES}})^\dagger \Gamma _n^{\mathrm{ES}}=\mathcal K_n\) implies injectivity of \(\Gamma _n^{\mathrm{ES}}\). The inverse-Gram identities of Theorem 11.4.18 identify \(S_{\Gamma _n^{\mathrm{ES}}}\) with \(\mathcal K_n^{-1}\), so the pointwise bounds of Theorem 11.4.19 apply unchanged.
Let \(c{\gt}0\), let \(0{\lt}r{\lt}1\), and let \(l,K\in \mathbb N\). If \(cr^l{\lt}1\), then
Since \(0{\lt}r{\lt}1\), the sequence \(r^m\) is nonincreasing. Each of \(l+1\), \(K+l\), and \(K+l+1\) is at least \(l\), so every displayed inequality follows from \(cr^l{\lt}1\).
Let \(A\) be a primitive MPS tensor whose fixed-point witness \(\rho \) is positive definite, and put \(\mathcal K_\infty =\mathscr R(P_\rho )\). For every \(0{\lt}a{\lt}1\), all sufficiently large \(n\) satisfy
The positive-definite fixed point makes \(\mathcal K_\infty \) invertible. Gram convergence implies that eventually
Factoring \(\mathcal K_n\) around the invertible base \(\mathcal K_\infty \) and applying the arbitrary-base Neumann estimate gives (172).
Under the same primitive-MPS and positive-definite fixed-point hypotheses,
in operator norm.
Inversion is continuous at the invertible operator \(\mathcal K_\infty \), so the claim follows from \(\mathcal K_n\to \mathcal K_\infty \).
Under the same hypotheses, for every \(0{\lt}a{\lt}1\) and all sufficiently large \(n\), the physical boundary map \(\Gamma _n^{\mathrm{ES}}\) is injective and its inverse Gram operator is \(\mathcal K_n^{-1}\), satisfying
Eventual invertibility of \((\Gamma _n^{\mathrm{ES}})^\dagger \Gamma _n^{\mathrm{ES}}\) implies eventual injectivity of \(\Gamma _n^{\mathrm{ES}}\). The inverse-Gram identities of Theorem 11.4.18 then identify \(S_{\Gamma _n^{\mathrm{ES}}}\) with \(\mathcal K_n^{-1}\), and the estimate of Theorem 11.4.22 applies.
For injective \(T\), define the operator \(Q_T:=T S_T T^\dagger \) on its codomain.
If \(T\) is an injective map from a finite-dimensional Hilbert space into a complete Hilbert space, then
The vector \(Q_Tx\) lies in \(\operatorname{range}T\). Since \(T^\dagger Q_Tx=(T^\dagger T)S_TT^\dagger x=T^\dagger x\), the adjoint identity gives \(\langle x-Q_Tx,Ty\rangle =0\) for every \(y\). Thus \(x-Q_Tx\) is orthogonal to \(\operatorname{range}T\), and the range–orthogonal decomposition characterizing the orthogonal projector gives \(Q_T=P_{\operatorname{range}T}\).
If \(A\) is \(L\)-block injective, then the injective range projector of \(\Gamma _L^{\mathrm{ES}}\) is the physical orthogonal projector onto the local ground space:
In particular, this is the physical range projector on the local Hilbert space, not the virtual transfer fixed-point projector.
The general injective range-projector theorem identifies \(Q_{\Gamma _L^{\mathrm{ES}}}\) with the orthogonal projector onto \(\operatorname{range}(\Gamma _L^{\mathrm{ES}})\). The exact-range theorem identifies this range with \(G_L^{\mathrm{ES}}(A)\).
Suppose that \(L_0{\gt}0\), that \(A\) is \(L_0\)-block injective, and that \(L\ge L_0\). Then the physical orthogonal projector onto the local ground space is
Thus the displayed operator is the physical range projector on \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\).
Positive-length propagation makes \(A\) block injective at length \(L\). Apply the physical range-projector identity at that length and unfold its inverse-Gram expression.
If \(\lVert t\rVert \le a{\lt}1\) in a normed ring with convergent geometric series and \(\lVert 1\rVert =1\), then
Since \(\lVert t\rVert \le a{\lt}1\), the Neumann series converges and \((1-t)^{-1}=\sum _{n\ge 0}t^n\). The triangle inequality and submultiplicativity give
where the last step uses monotonicity of the reciprocal on the positive reals.
Let \(\mathbb K\) be a real or complex scalar field, let \(E\) be a finite-dimensional complete \(\mathbb K\)-Hilbert space, and let \(F\) be a complete \(\mathbb K\)-Hilbert space. Let \(T\colon E\to F\) be an injective continuous linear map. If \(a\in \mathbb R\) satisfies \(a{\lt}1\) and
then
If \(E\) is trivial, then \(S_T=0\) and the bound follows from \(a{\lt}1\). Otherwise, identify \(S_T\) with the ring inverse of \(T^\dagger T\) and apply Theorem 11.4.29 to \(t=I_E-T^\dagger T\).
The parent interaction at range \(L\) is the canonical orthogonal projector specified in (14).
The local parent interaction is a projector: \(h_L(A)^2=h_L(A)\).
This is the idempotency of the orthogonal projector onto \(G_L(A)^\perp \), transported from the Hilbert-space realization of the local coefficient space.
For any \(v\in G_L(A)\), one has \(h_L(A)v=0\).
Since \(h_L(A)\) is the orthogonal projector onto \(G_L(A)^\perp \), it annihilates every element of \(G_L(A)\).
For \(\sigma \in \{ 0,\ldots ,d{-}1\} ^{N}\) and a starting site \(i\), write \(\sigma |_{[i,i+L-1]}\in \{ 0,\ldots ,d{-}1\} ^{L}\) for the word defined by \((\sigma |_{[i,i+L-1]})(r)=\sigma (i+r\bmod N)\) for \(0\le r{\lt}L\).
If \(L\le N\) and \(\tau \in \{ 0,\ldots ,d{-}1\} ^{L}\), write \(\sigma ^{[i,i+L-1]\leftarrow \tau }\) for the configuration defined by
The index \(r\) in the first case is unique because \(L\le N\).
If \(L\le N\), then restricting to the same cyclic interval after prescribing its values recovers the inserted word: \(\left(\sigma ^{[i,i+L-1]\leftarrow \tau }\right)|_{[i,i+L-1]}=\tau \).
If \(L\le N\), then prescribing on a cyclic interval the values already carried by \(\sigma \) leaves \(\sigma \) unchanged: \(\sigma ^{[i,i+L-1]\leftarrow \sigma |_{[i,i+L-1]}}=\sigma \).
Case-split on whether the offset \((k-i+N)\bmod N\) lies in \([0,L)\): inside the cyclic interval the prescribed value is the original value of \(\sigma \); outside the interval both sides already agree with \(\sigma \).
If \(L\le N\), then two prescriptions on the same cyclic interval reduce to the second prescription:
On the cyclic interval both sides take the value prescribed by \(\upsilon \). Away from that interval neither second prescription changes the value.
For a periodic chain of length \(N\), the parent Hamiltonian is the sum of translated copies of the local interaction.
For each site index \(i\in \{ 0,\ldots ,N{-}1\} \), the translated local term is determined by
If \(L\le N\), then the translated term is obtained by applying the local parent interaction on the chosen cyclic window:
This is the first case in (187).
For every site \(i\), one has \(h_i(A,L)^2=h_i(A,L)\).
If \(L{\gt}N\), the translated term is zero. Suppose \(L\le N\) and put \(f_{\sigma ,i}(\tau ) :=\psi (\sigma ^{[i,i+L-1]\leftarrow \tau })\). The pointwise formula for translated terms, together with Lemmas 11.4.36 and 11.4.38, gives
using \(h_L(A)^2=h_L(A)\) by Lemma 11.4.32.
The chain Hamiltonian is
A vector \(\psi \) is a frustration-free ground state of the parent Hamiltonian when each local interaction annihilates it, that is, when \(h_i\psi =0\) for every \(i\in \{ 0,\ldots ,N{-}1\} \).
For any window of \(L\) consecutive sites starting at position \(i\) in an \(N\)-site periodic chain with \(L\le N\), the restriction of the periodic MPS vector \(V^{(N)}(A)\) to that window lies in \(G_L(A)\).
The boundary matrix is the product of the \(A\)-matrices on the complement sites. Trace cyclicity rotates the full \(N\)-site product so that the window indices come first, matching the definition of the ground-space map.
For each site \(i\), one has \(h_iV^{(N)}(A)=0\).
This combines periodic-MPS-vector window membership with the fact that the parent interaction annihilates ground-space elements.
For every \(N\) with \(L\le N\), the periodic MPS vector satisfies \(H_N(A,L)V^{(N)}(A)=0\).
This is a sum of zeros, since each local term annihilates the periodic MPS vector.
The periodic MPS vector is a frustration-free ground state: for every site \(i\), one has \(h_iV^{(N)}(A)=0\).
The claim follows from Lemma 11.4.45.
11.5 Intersection property and unique ground state
11.5.1 Restriction maps
Given a state \(\psi \) on \(L+1\) sites, we define two restriction maps.
Appending a last letter to a word corresponds to right multiplication by the corresponding tensor letter, while prepending a first letter corresponds to left multiplication. The restriction maps are the associated linear maps on the state spaces under the computational-basis identification, together with their pointwise formulas.
Fixing the last physical index \(j\) gives the state \(\sigma \mapsto \psi (\sigma _1,\ldots ,\sigma _L,j)\).
Fixing the first physical index \(i\) gives the state \(\sigma \mapsto \psi (i,\sigma _1,\ldots ,\sigma _L)\).
A state \(\psi \) on \(L+1\) sites satisfies the left ground condition if, for every \(j\), the state obtained by fixing the last index to \(j\) lies in \(G_L(A)\).
A state \(\psi \) on \(L+1\) sites satisfies the right ground condition if, for every \(i\), the state obtained by fixing the first index to \(i\) lies in \(G_L(A)\).
11.5.2 Forward direction: ground space restricts
If \(\psi \in G_{L+1}(A)\), i.e. if \(\psi (\sigma )=\operatorname{tr}(A^\sigma X)\) for some \(X\), then, for each \(j\), the state obtained by fixing the last index to \(j\) lies in \(G_L(A)\) with witness \(A^jX\).
The restriction simply absorbs the last tensor into the boundary matrix:
![\begin{tenkz}[rows={wire}, cols=5, boundary=periodic]
\tn[skin=ring, label pos=90]{A^j X} &
\tn[label pos=135, ports={90:physical:$\sigma_1$}]{A} &
\tn[label pos=135, ports={90:physical}]{A} & \tn[skin=dots]{} &
\tn[label pos=135, ports={90:physical:$\sigma_L$}]{A}
\tnmark[form=bracket]{(1,2) .. (1,5)}{$L$}
\end{tenkz}](tenkz_svg/tenkz-8ef474137f08ef9b.svg)
Direct computation gives
If \(\psi \in G_{L+1}(A)\) with \(\psi (\sigma )=\operatorname{tr}(A^\sigma X)\), then, for each \(i\), the state obtained by fixing the first index to \(i\) lies in \(G_L(A)\) with witness \(XA^i\).
On the right restriction, one first rotates the trace and then reads the resulting boundary matrix on the shortened window:
![\begin{tenkz}[rows={wire}, cols=5, boundary=periodic]
\tn[skin=ring, label pos=90]{X A^i} &
\tn[label pos=135, ports={90:physical:$\sigma_1$}]{A} &
\tn[label pos=135, ports={90:physical}]{A} & \tn[skin=dots]{} &
\tn[label pos=135, ports={90:physical:$\sigma_L$}]{A}
\tnmark[form=bracket]{(1,2) .. (1,5)}{$L$}
\end{tenkz}](tenkz_svg/tenkz-0be8ffd053fc736d.svg)
By trace cyclicity,
11.5.3 Injectivity of the ground-space map
If \(A\) is injective and \(L \ge 1\), then \(\Gamma _L\) is injective.
If \(\Gamma _L(X) = 0\), then \(\operatorname{tr}(A^\sigma X) = 0\) for every length-\(L\) word \(\sigma \). Since \(A\) is injective, the products \(\{ A^\sigma \} _\sigma \) span \(M_{D}(\mathbb {C})\), so the trace pairing gives \(X = 0\).
If the length-\(L\) products \(\{ A^\sigma : |\sigma | = L\} \) span \(M_{D}(\mathbb {C})\), then \(\Gamma _L\) is injective.
If \(\Gamma _L(X) = 0\), then \(\operatorname{tr}(A^\sigma X) = 0\) for every word \(\sigma \) of length \(L\). The spanning hypothesis turns this into \(\operatorname{tr}(MX) = 0\) for every \(M \in M_{D}(\mathbb {C})\), and the trace pairing gives \(X = 0\).
If \(A\) is \(L_0\)-block injective, then \(\Gamma _{L_0}\) is injective.
If \(A\) is injective and \(L \ge 1\), then \(\dim G_L(A) = D^2\).
The upper bound \(\dim G_L(A) \le D^2\) is Lemma 11.2.1.9. Injectivity of \(\Gamma _L\) gives \(\dim G_L(A) = \dim M_{D}(\mathbb {C}) = D^2\).
11.5.4 The intersection property
The following two statements hold for every tensor \(A\); injectivity enters only through the choice of the matrices \(R_j\) in the second one.
Let \(\psi \) be a vector on \(K+2\) sites and let \(Y_i,Z_j\in M_{D}(\mathbb {C})\) satisfy \(\psi (i,\tau )=\Gamma _{K+1}(Y_i)(\tau )\) and \(\psi (\tau ,j)=\Gamma _{K+1}(Z_j)(\tau )\) for all letters \(i,j\) and all words \(\tau \) of length \(K+1\). Then
for all letters \(i,j\).
For a word \(\sigma \) of length \(K\), cyclicity of the trace gives
Let \(\psi \) be a vector on \(K+1\) sites and let \(Y_i,Z_j\in M_{D}(\mathbb {C})\) satisfy \(\psi (i,\tau )=\Gamma _{K}(Y_i)(\tau )\) for all letters \(i\) and words \(\tau \) of length \(K\), and \(A^jY_i=Z_jA^i\) for all letters \(i,j\). If matrices \(R_j\in M_{D}(\mathbb {C})\) satisfy \(\sum _jR_jA^j=\mathbb {1}\), then \(\psi \in G_{K+1}(A)\).
Put \(X=\sum _jR_jZ_j\). Then
so \(\psi =\Gamma _{K+1}(X)\).
If \(A\) is injective and \(L \ge 2\), then the nontrivial inclusion in the intersection property holds:
Comparing the left and right restrictions on their common \((L-1)\)-site overlap determines a boundary matrix \(X'\) that reconstructs the full \((L+1)\)-site word:
![\begin{tenkz}[rows={wire}, cols=5, boundary=periodic]
\tn[skin=ring]{X'} &
\tn[label pos=s, ports={90:physical:$\sigma_1$}]{A} &
\tn[label pos=s, ports={90:physical}]{A} & \tn[skin=dots]{} &
\tn[label pos=s, ports={90:physical:$\sigma_{L+1}$}]{A}
\tnmark[form=bracket]{(1,2) .. (1,5)}{$L+1\text{ sites}$}
\end{tenkz}](tenkz_svg/tenkz-d62d3ecf71aff8c7.svg)
From the right ground condition, for each physical index \(i\) there exists a unique \(Y_i \in M_{D}(\mathbb {C})\), by injectivity of \(\Gamma _L\), such that \(\psi (i,\sigma ) = \operatorname{tr}(A^\sigma Y_i)\).
Similarly, from the left ground condition, for each \(j\) there exists a unique \(Z_j\) such that \(\psi (\sigma ,j) = \operatorname{tr}(A^\sigma Z_j)\).
Matching on the \((L-1)\)-site overlap (Theorem 11.5.4.1) and using injectivity of \(\Gamma _{L-1}\) gives \(A^jY_i = Z_jA^i\) for all \(i,j\).
Since \(A\) is injective, the matrices \(\{ A^j\} \) span \(M_{D}(\mathbb {C})\) (Definition 2.5.1), so \(\mathbb {1}= \sum _j c_jA^j\) for some coefficients \(c_j \in \mathbb {C}\). Thus, with \(X' := \sum _j c_jZ_j\), \(Y_i = \mathbb {1}Y_i = \sum _j c_jA^jY_i = \sum _j c_jZ_jA^i = X'A^i\). This is Theorem 11.5.4.2 with \(R_j=c_j\mathbb {1}\).
By trace cyclicity, \(\psi (\sigma ) = \operatorname{tr}(A^\sigma X')\), so \(\psi \in G_{L+1}(A)\).
For injective \(A\) and \(L \ge 2\), \(G_L^{\mathrm{left}}\cap G_L^{\mathrm{right}} = G_{L+1}(A)\), equivalently, \(\psi \in G_{L+1}(A)\) if and only if \(\psi \in G_L^{\mathrm{left}}\cap G_L^{\mathrm{right}}\).
The claim follows from the two forward-direction lemmas and Theorem 11.5.4.3.
11.5.5 Unique ground state
Contiguous and cyclic interval restrictions
For a non-wrapping interval \([s,s+M-1] \subseteq \{ 0,\ldots ,N-1\} \) and a full configuration \(\rho \) supplying the complementary values, the restriction map is
These identities include the boundary cases \(s=0\) and \(M=N\), the deletion of the first or last site, and the composition of fixing \(K\) prefix sites followed by restricting to \([s+K,s+K+L-1]\).
On the periodic chain, for a full configuration \(\rho \) supplying the complementary values, put
The restriction to the cyclic interval beginning at \(i\) is
with all site labels read modulo \(N\). If \(L\le N\), this is the same configuration as \(\rho ^{[i,i+L-1]\leftarrow \omega }\). If two boundary conditions agree whenever \(\delta _i(k)\ge L\), their restrictions are equal. In particular, filling the interval inside the outside configuration leaves the restriction unchanged:
For every \(r{\lt}L\), the inserted word is recovered as \(\rho ^{\mathrm{cyc}}_{i,L,\omega }(i+r)=\omega (r)\) with site labels taken modulo \(N\). Reconstructing a configuration from its local restriction returns the original configuration: taking \(\rho =\sigma \) and \(\omega (r)=\sigma (i+r)\) for \(r{\lt}L\),
Modifying a single inserted site commutes with the cyclic extension: for \(r{\lt}L\) and a letter \(x\), writing \(\omega [r\mapsto x]\) for the word \(\omega \) with its value at \(r\) replaced by \(x\),
Deleting the final site of an \((L+1)\)-interval gives the \(L\)-interval beginning at \(i\); for \(L+1\le N\), deleting the first site gives the \(L\)-interval beginning at \(i+1\); and the cyclic formula agrees with the contiguous restriction when the interval does not wrap around the ring.
If a cyclic \((L+1)\)-window is represented by a boundary matrix \(Y\), then deleting one endpoint gives
Hence two adjacent cyclic windows whose restrictions agree on the common length-\(L\) overlap have boundary matrices satisfying \(Y_1A^a=A^bY_2\). Consequently, for a finite chain of adjacent overlaps indexed by \(0\le r{\lt}n\), satisfying \(Y_rA^{a_r}=A^{b_r}Y_{r+1}\), one has \(Y_0A^{a_0}\cdots A^{a_{n-1}} = A^{b_0}\cdots A^{b_{n-1}}Y_n\). In particular, when the boundary words are given by \(a_r=\rho (i_0+r)\) and \(b_r=\rho (i_0+r+L+1)\) modulo \(N\), this identity reads \(Y_0A^a=A^bY_n\).
The first two identities are the cyclic first- and last-site deletion identities, followed by the corresponding ground-space restriction formulas. For adjacent windows, equality of the completed outside configurations gives \(\Gamma _L(Y_0A^a)=\Gamma _L(A^bY_n)\). Injectivity of \(\Gamma _L\) gives \(Y_0A^a=A^bY_n\). Iterating the adjacent identity gives the word-product identity.
If an \(N\)-site state satisfies the \(L\)-site ground condition on every non-wrapping contiguous window, with \(L \ge 2\), then it lies in \(G_N(A)\).
The induction repeatedly merges two neighbouring admissible windows \([i,i+L-1]\) and \([i+1,i+L]\) into the larger window \([i,i+L]\) by applying Theorem 11.5.4.3 to their common overlap \([i+1,i+L-1]\).
Fix a prefix \(u\in \{ 0,\ldots ,d-1\} ^K\). The suffix restriction of an \((K+L)\)-site state \(\psi \) is the \(L\)-site state given by \((R^{\mathrm{tail}}_u\psi )(\sigma )=\psi (u,\sigma )\).
A state \(\psi \) on \(K+L\) sites lies in the suffix ground condition if every fixed prefix \(u\) gives a suffix state in \(G_L(A)\).
If \(\psi \in G_{K+L}(A)\), then every fixed-prefix suffix restriction of \(\psi \) lies in \(G_L(A)\). For a vector in \(G_{L+1}(A)\), fixing either endpoint gives the expected left or right multiplication of its boundary matrix.
Write \(\psi (\rho ) = \operatorname{tr}(A^\rho X)\) for a boundary matrix \(X\). Fixing the final site gives the boundary matrix \(A^jX\), while fixing the first site gives \(XA^i\) by cyclicity of the trace. Fixing a prefix \(u\) gives \(\psi (u,\sigma )=\operatorname{tr}(A^uA^\sigma X)=\operatorname{tr}(A^\sigma XA^u)\), so the suffix restriction again has the form of a ground-space vector.
The two-sided window restriction is recorded in the martingale chapter as Theorem 12.3.7.13: for sites \(s,L,N\) with \(s+L \le N\), a full configuration \(\tau \) supplying the complementary values, and a boundary matrix \(X\), the nonwrapping restriction of the open-boundary ground-space vector \(\Gamma _N(X)\) to the window \([s,s+L-1]\) is again an open-boundary ground-space vector,
Both outside words are absorbed into the boundary matrix: writing \(w_L=(\tau _0,\ldots ,\tau _{s-1})\) and \(w_R=(\tau _{s+L},\ldots ,\tau _{N-1})\), word multiplicativity and cyclicity of the trace give, for any window word \(\omega \),
so the restriction is \(\Gamma _L(A^{w_R}XA^{w_L})\). This two-sided absorption generalizes Lemma 11.5.5.7, whose suffix restriction absorbs only the prefix.
Assume \(\psi \) on \(K+L_0+1\) sites satisfies the left ground condition on the first \(K+L_0\) sites and the suffix ground condition on every fixed prefix of length \(K\). Then there exist matrices \((Z_j)_j\) and \((Y_u)_u\) with the two trace representations below and, for all \(j\) and \(u\), the compatibility relation
Choose \(Z_j\) from the long left-window condition and \(Y_u\) from the suffix ground condition. The two restrictions give the same vector in \(G_{L_0}(A)\), namely \(\Gamma _{L_0}(Z_jA^u)=\Gamma _{L_0}(A^jY_u)\). Injectivity of \(\Gamma _{L_0}\), obtained from \(L_0\)-block injectivity, gives \(Z_jA^u=A^jY_u\).
Assume \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\). If matrices \((Z_j)_j\) satisfy \(Z_jA^\sigma =A^jY_\sigma \) for every word \(\sigma \) of length \(K\), then there exists a matrix \(X\) such that \(Z_j=A^jX\) for all \(j\).
For \(K = 1\), the hypothesis gives \(Z_jA^i=A^jY_i\) for every letter \(i\). Every blocked word of length \(L_0\) is nonempty, so this identity extends to the coefficients of the blocked tensor \(A^{[L_0]}\). Since \(A^{[L_0]}\) is injective, the identity matrix is a linear combination of those blocked coefficients, and substituting this decomposition yields \(Z_j=A^jX\) for a common matrix \(X\). For larger \(K\), strip the first letter: for each \(i\), the matrices \(Z_jA^i\) satisfy the same compatibility hypothesis with words of length \(K-1\), so the induction hypothesis reduces the problem to the case \(K=1\).
Assume \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\). If an element \(\psi \) on \(K+L_0+1\) sites satisfies the left ground condition on the first \(K+L_0\) sites and the suffix ground condition on every prefix of length \(K\), then \(\psi \in G_{K+L_0+1}(A)\).
Lemma 11.5.5.9 produces matrices \((Z_j)_j\) and \((Y_u)_u\) with \(Z_jA^u=A^jY_u\) for every prefix word \(u\) of length \(K\). Theorem 11.5.5.10 gives a common right factor \(X\) such that \(Z_j=A^jX\) for all \(j\). Therefore, each last-site restriction of \(\psi \) agrees with the corresponding last-site restriction of the ground-space vector determined by \(X\). Since every configuration is obtained by adjoining its final letter to its initial \((K+L_0)\)-tuple, the two states coincide, so \(\psi \) lies in \(G_{K+L_0+1}(A)\).
For every \(L\in \mathbb {N}\), let \(\mathcal U_L(A)\) be the subspace of the \((L+1)\)-site Hilbert space whose corresponding functions satisfy the left ground condition: every fixed-final-site restriction belongs to \(G_L(A)\). For every \(K,L\in \mathbb {N}\), let \(\mathcal V_{K,L}(A)\) be the subspace of the \((K+L)\)-site Hilbert space whose corresponding functions satisfy the tail ground condition: every fixed-prefix restriction belongs to \(G_L(A)\). These definitions include \(L=0\). Let \(P^{\mathrm{left}}_L(A)\) and \(P^{\mathrm{tail}}_{K,L}(A)\) be the orthogonal projectors onto \(\mathcal U_L(A)\) and \(\mathcal V_{K,L}(A)\), respectively.
Under the canonical \(\ell ^2\) identification, a Hilbert-space vector belongs to \(\mathcal U_L(A)\) exactly when the corresponding function satisfies the left ground condition: every fixed-final-site restriction belongs to \(G_L(A)\).
Transport through the canonical \(\ell ^2\) equivalence and unfold the left-site restriction condition.
Under the canonical \(\ell ^2\) identification, a Hilbert-space vector belongs to \(\mathcal V_{K,L}(A)\) exactly when the corresponding function satisfies the tail ground condition: every fixed-prefix restriction belongs to \(G_L(A)\).
Transport through the canonical \(\ell ^2\) equivalence and unfold the tail restriction condition.
If \(D\ge 1\), \(A\) is \(L_0\)-block-injective, and \(L_0{\gt}0\), then
In Nachtergaele’s notation, with \(n=K+L_0\) and \(l=L_0\), this is the common range identity for \(G_{\Lambda _n}\) and \(G_{\Lambda _{n+1}\setminus \Lambda _{n-l}}\) inside \(\Lambda _{n+1}\).
The forward inclusion is the growing-back step at the injectivity length. Conversely, a vector in the longer ground space remains in the ground space after every left or tail restriction.
Let \(A\) be \(L_0\)-block-injective with \(D \ge 1\) and \(L_0 {\gt} 0\). If an \(N\)-site state satisfies the ground-space constraint on every non-wrapping contiguous interval of length \(L_0+1\), for every fixed choice of the complementary sites, with \(N \ge L_0+1\), then it lies in \(G_N(A)\).
Induct on the extra length beyond \(L_0+1\). The induction hypothesis gives the long left-window ground condition after deleting the last site, while Definition 11.5.5.5 and Remark 11.5.5.1 identify every fixed-prefix suffix with one of the assumed length-\(L_0+1\) intervals. Theorem 11.5.5.11 then grows back the rightmost site.
Let \(S_m\subseteq (\mathbb {C}^d)^{\otimes m}\) be subspaces, and let \(0{\lt}L\le N\). Suppose that every non-wrapping length-\(L\) interval of \(\psi \), for every fixed choice of the complementary sites, lies in \(S_L\). Suppose also that, for every \(m\ge L\), \((\mathbb {C}^d\otimes S_m)\cap (S_m\otimes \mathbb {C}^d) = S_{m+1}\). Then \(\psi \in S_N\).
Induct on the length \(m\) of a non-wrapping interval. The case \(m=L\) is the assumed local condition. If all length-\(m\) intervals lie in \(S_m\), then the two one-site restrictions of a length-\((m+1)\) interval are adjacent length-\(m\) intervals. Hence the length-\((m+1)\) interval lies in \((\mathbb {C}^d\otimes S_m)\cap (S_m\otimes \mathbb {C}^d)\), and the displayed identity places it in \(S_{m+1}\). Taking \(m=N\) and the interval starting at the first site gives \(\psi \in S_N\).
Let \(S_m\subseteq (\mathbb {C}^d)^{\otimes m}\) be subspaces, and let \(0{\lt}L\le N\). Suppose that \(G_L(A)\subseteq S_L\). Suppose also that, for every \(m\ge L\), \((\mathbb {C}^d\otimes S_m)\cap (S_m\otimes \mathbb {C}^d) = S_{m+1}\). Then the periodic local constraints imply \(\mathcal G_{N,L}(A)\subseteq S_N\).
If \(\psi \in \mathcal G_{N,L}(A)\), then every length-\(L\) cyclic interval of \(\psi \) lies in \(G_L(A)\), and hence in \(S_L\). A non-wrapping interval is the cyclic interval starting at the same site. Therefore, all non-wrapping length-\(L\) intervals lie in \(S_L\), and Lemma 11.5.5.17 gives \(\psi \in S_N\).
On a nonempty periodic chain, longer cyclic-window constraints imply all shorter cyclic-window constraints: if \(L'\le L\le N\), then \(\mathcal G_{N,L}(A)\subseteq \mathcal G_{N,L'}(A)\).
For every cyclic window vector \(v\in G_{L+1}(A)\) and every last letter \(b\), one has \(v|_{\mathrm{last}=b}\in G_L(A)\). Thus \(\mathcal G_{N,L+1}(A)\subseteq \mathcal G_{N,L}(A)\), and iteration gives the general antitonicity.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D \ge 1\). If an \(N\)-site periodic-chain state satisfies all cyclic ground-space constraints at some range \(L_0+1\le L\le N\), then it lies in the open-chain ground space \(G_N(A)\).
11.6 Transport under physical blocking
For a tensor \(A=(A^i)_{i=0}^{d-1}\) and a block length \(p\), write \(A^{[p]}\) for the tensor whose physical matrices are the consecutive products \(A^{i_1}\cdots A^{i_p}\). We distinguish the number \(N\) of blocked sites from the number \(Np\) of original sites.
Grouping each consecutive \(p\)-tuple of physical indices defines a canonical computational-basis unitary satisfying
Its forward and inverse coordinate formulas are the corresponding reindexings by the blocked-configuration equivalence and its inverse.
For every \(x\in \ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\), the Euclidean boundary maps intertwine with the blocked-configuration isometry:
For a blocked configuration \(J=(J_1,\ldots ,J_N)\), its flattened word \(\sigma \), and a boundary matrix \(X\in M_{D}(\mathbb {C})\), blocked-word evaluation gives
Applying the Frobenius transport formula to the coefficient identity yields (192).
Let \(A=(A^i)\) and \(C=(C^i)\) have the same physical alphabet, and suppose there is a coisometry \(U:\mathbb {C}^D\to \mathbb {C}^n\) such that \(UU^\dagger =I_n\) and \(A^i=U^\dagger C^iU\) for every \(i\). Then, for every \(L\geq 1\),
In particular, if \((A_j)_{j=1}^g\) are the distinct BNT representatives selected from literal canonical-form data for \(A\), then, for \(p,L\geq 1\),
Identity (195) is a derived trace identity for an exact coisometric reconstruction. It is not the literal direct-sum canonical-form equation of CPSV16.
For every word \(w\) of positive length, \(A^w=U^\dagger C^wU\). Cyclicity of the trace gives
where (197) uses \(UU^\dagger =I_n\). This proves (195). For literal canonical-form data, apply this equality to the retained direct sum. Theorem 13.3.6 gives the sum over all displayed normal blocks. Blocks in one BNT phase class differ by a nonzero phase and an invertible gauge, so their local spaces agree; the sum therefore reduces to the distinct representatives, which proves (196).
Let \(B=\bigoplus _{j=1}^{g}\mu _jA_j\), where every \(\mu _j\) is nonzero and every bond dimension is positive. Suppose that the simultaneous one-letter evaluations span the product algebra:
Then, for every \(N{\gt}2\),
Equation (198) supplies common coefficients \(c_i\) satisfying \(\sum _i c_iA_j^i=I_{D_j}\) for every \(j\). In the restriction-intersection argument of the proof of [ 228 , Theorem 12 ] , these coefficients replace the right-canonical identity and yield (199). This conditional variant and its relation to the source proof are documented in [ 40 ] . The same span makes every \(A_j\) injective, so Theorem 11.11.2.1 identifies its periodic chain space with \(\operatorname{span}\{ |V^{(N)}(A_j)\rangle \} \). The block-diagonal comparison of Lemma 12.1.25 then gives (200).
Let \(B\) and \(C\) have the same physical alphabet and satisfy \(G_L(B)=G_L(C)\). Their periodic chain spaces agree for every \(N\):
Their parent-Hamiltonian kernels agree for every \(N{\gt}L\):
Consequently, any fixed BNT-vector family satisfying the parent-Hamiltonian spanning equation for \(C\) also satisfies it for \(B\).
When \(N=0\) or \(L{\gt}N\), both chain spaces are \(\top \) by definition. Otherwise \(0{\lt}N\) and \(L\leq N\). Every translated local constraint is then the inverse image of the same subspace \(G_L(B)=G_L(C)\), so their intersections are equal. This gives (201) for every \(N\). When \(N{\gt}L\), Theorem 12.3.17 identifies those intersections with the two parent-Hamiltonian kernels and gives (202).
Let \(A\) be in literal CPSV canonical form with positive ambient bond dimension \(D\), and suppose that the physical dimension satisfies \(1{\lt}d\). There are a basis of normal tensors \((A_j)_{j=1}^{g}\) and a positive integer \(L\) such that
and, for every original periodic chain of length \(N{\gt}L\),
Thus the orthogonal complement of the local MPS space is nonzero, and the corresponding bounded-range interaction is a parent Hamiltonian. This is the construction of [ 13 , Definition 3.9 and line 527 ] , with its condition \(d^L{\gt}D^2\) made explicit.
Choose one normal representative from each BNT phase class and put each representative in a Perron gauge for which
with \(\Lambda _j\) positive definite. Their dimensions satisfy \(\sum _jD_j\leq D\), so \(g\leq D\), and quantum Wielandt makes every representative block-injective at the common length \(L_0=D^4\).
Suppose first that \(g\geq 2\). Put
If \(N{\gt}L_{\mathrm{aux}}\), then \(N\geq R+L_0\). The source-normalized form of [ 228 , Theorem 12 ] therefore identifies the range-\(R\) parent-Hamiltonian kernel with the BNT span. Since \(R\leq L_{\mathrm{aux}}\), cyclic-window antitonicity and the usual frustration-free inclusion give
Thus the range-\(L_{\mathrm{aux}}\) kernel is exactly the BNT span for every \(N{\gt}L_{\mathrm{aux}}\), and
If \(g=1\), the same spanning conclusion holds at \(L_{\mathrm{aux}}=D^4+1\) by the normal-tensor intersection and closure property reviewed in [ 17 , Section IV.C ] . If \(g=0\), the one-site block-diagonal spanning result gives it at \(L_{\mathrm{aux}}=2\); this is a formal boundary extension of the source argument.
In every case enlarge the interaction range to the common value \(L=3D^5\). For every \(N{\gt}L\), range antitonicity and the frustration-free inclusion give the second kernel sandwich
Hence the all-\(N{\gt}L\) spanning equation is preserved. Since \(D{\gt}0\) and \(d\geq 2\),
Exact coisometric reconstruction identifies the local space of the representative direct sum with \(G_L(A)\) and transfers the equality to the original tensor. The chosen range is positive and satisfies \(L\leq 3D^5\) by equality.
Let \(A\) be in literal CPSV canonical form with positive ambient bond dimension \(D\). There are a basis of normal tensors \((A_j)_{j=1}^{g}\) and an integer \(p\) satisfying \(1\leq p\leq 3D^5\) such that \((A_j^{[p]})_{j=1}^{g}\) is a basis of normal tensors for \(A^{[p]}\) and, for every \(N{\gt}2\),
Thus the actual blocked tensor has a nearest-neighbor parent Hamiltonian on the blocked lattice. This is distinct from Theorem 11.6.6, which supplies a nonzero bounded-range interaction and the parent-Hamiltonian spanning identity on every sufficiently long original chain. The two conclusions and the argument closing the short-ring window are recorded in [ 34 ] .
Choose \(p\) so that the simultaneous one-letter evaluations of the blocked normal representatives span the product of their full matrix algebras. Positive blocking preserves the BNT property. Set \(C^i=\bigoplus _{j=1}^{g}(A_j^{[p]})^i\). Then Theorem 11.6.4 gives, for every \(N{\gt}2\),
After the repeated phase-gauge copies and the ambient coisometry are removed, Lemma 11.6.3 gives
Lemma 11.6.5 identifies the two parent-Hamiltonian kernels, so (206) becomes (205).
The Euclidean local space of the blocked tensor is carried exactly to the original Euclidean local space:
Hence their orthogonal projectors satisfy
These statements include \(N=0\) or \(p=0\); in those cases the same explicit configuration isometry and boundary-map calculation apply. In particular, the projector identity holds for three blocked sites and \(3p\) original sites. These identities concern the full local ground spaces and their orthogonal projectors under blocking.
The exact range identification for the Euclidean boundary map and (192) give
which is (208). For every subspace \(K\), orthogonal projection under the unitary \(W_{N,p}\) satisfies
Taking \(K=G_N^{\mathrm{ES}}(A^{[p]})\) gives (209).
11.7 Ground-space correspondence under an invertible on-site map
An invertible map \(\Lambda \) on the physical space \(\mathbb {C}^d\) acts on a tensor by mixing its physical matrices, \(A^i\mapsto \sum _j\Lambda _{ij}A^j\) [ 17 , Section IV.C ] . Under this action, the deformed local interaction \(h'=({\Lambda ^{-1}}^\dagger )^{\otimes L}h(\Lambda ^{-1})^{\otimes L}\) is a parent interaction for the deformed tensor, and the ground spaces of the two chain Hamiltonians correspond one-to-one through \(\Lambda ^{\otimes N}\).
Scope restriction (periodic-chain specialization): The ground-space correspondence under physical deformation is stated in [ 17 , Section IV.C ] for tensor network states on a general graph; here it is formalized for matrix product states on the periodic chain. Documented in [ 104 ] .
For a matrix \(\Lambda \in M_{d}(\mathbb {C})\) and a tensor \(A=(A^i)_{i=0}^{d-1}\), the physically rotated tensor \(\Lambda A\) has matrices
Rotation by the identity is trivial: \(\mathbb {1}A=A\).
Since \(\mathbb {1}_{ij}=\delta _{ij}\), each component in (212) reduces to \(\sum _j\delta _{ij}A^j=A^i\).
For \(\Lambda ,\Lambda '\in M_{d}(\mathbb {C})\),
For \(\Lambda \in M_{d}(\mathbb {C})\) and \(L\ge 0\), the tensor power \(\Lambda ^{\otimes L}\) is the matrix on \(L\)-site configurations with entries
It acts on the \(L\)-site space \(\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C}\) by matrix multiplication and on its Hilbert-space realization \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\) in the same way.
For \(\Lambda ,\Lambda '\in M_{d}(\mathbb {C})\) one has \((\Lambda \Lambda ')^{\otimes L}=\Lambda ^{\otimes L}\Lambda '^{\otimes L}\), \(\mathbb {1}^{\otimes L}=\mathbb {1}\), and \((\Lambda ^{\otimes L})^\dagger =(\Lambda ^\dagger )^{\otimes L}\). In particular, \(\Lambda ^{\otimes L}\) is unitary when \(\Lambda \) is. If \(\Lambda \) is invertible, then \((\Lambda ^{-1})^{\otimes L}\) is a two-sided inverse of \(\Lambda ^{\otimes L}\).
Reading the blocked physical index of \(L\) consecutive sites as an \(L\)-site configuration identifies \(\Lambda ^{\otimes L}\) with the Kronecker lift of \(\Lambda \) through blocking, for which multiplicativity, the identity, and the conjugate transpose are known. The inverse statements follow from multiplicativity applied to \(\Lambda ^{-1}\Lambda =\Lambda \Lambda ^{-1}=\mathbb {1}\).
Let \(\Lambda :\mathbb C^d\to \mathbb C^m\) be a linear map. For every word \(\sigma \) of length \(N\) in the \(m\)-letter alphabet,
where \(\tau \) ranges over words in the \(d\)-letter alphabet.
Expand each factor of \((\Lambda A)^{\sigma _1}\cdots (\Lambda A)^{\sigma _N}\) by (212) and collect the coefficient of \(A^{\tau }\), by induction on \(N\).
For \(\Lambda \in M_{d}(\mathbb {C})\), \(|V^{(N)}(\Lambda A)\rangle =\Lambda ^{\otimes N}|V^{(N)}(A)\rangle \) and \(\Gamma _L^{\Lambda A}(X)=\Lambda ^{\otimes L}\, \Gamma _L^{A}(X)\) for every boundary matrix \(X\in M_{D}(\mathbb {C})\).
For every \(\Lambda \in M_{d}(\mathbb {C})\) and \(L\ge 0\),
and the same identity holds for the Hilbert-space realizations of the two local ground spaces.
Apply Theorem 10.4.9.30 to the square map \(\Lambda \). Its tensor power agrees with the on-site tensor power in the configuration basis.
Following [ 17 , Section IV.C, Definition (Parent Hamiltonian) ] , a parent interaction on \(L\) sites for the tensor \(A\) is a hermitian positive semidefinite operator \(h\ge 0\) on \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\) whose kernel equals \(G_L(A)\).
The canonical parent interaction \(P_L^{\mathrm{ES}}(A)\), the orthogonal projector onto \(G_L(A)^\perp \), is a parent interaction on \(L\) sites for \(A\).
For \(\Lambda \in M_{d}(\mathbb {C})\) and an operator \(h\) on \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\), the deformed interaction is
This is the operator written in [ 17 , Section IV.C ] as \(h'=(A^{-1}\otimes A^{-1})^\dagger h(A^{-1}\otimes A^{-1})\) for the two-site interaction of an isometric tensor network, and as \(h'=({\Lambda ^{-1}}^\dagger )^{\otimes k}h(\Lambda ^{-1})^{\otimes k}\) for a general physical perturbation.
Let \(\Lambda \in M_{d}(\mathbb {C})\) be invertible and let \(h\) be a parent interaction on \(L\) sites for \(A\). Then the deformed interaction \(h'\) of (217) is a parent interaction on \(L\) sites for \(\Lambda A\): \(h'\ge 0\) and \(\ker h'=G_L(\Lambda A)\).
Write \(S=(\Lambda ^{-1})^{\otimes L}\). By Lemma 11.7.5, \(({\Lambda ^{-1}}^\dagger )^{\otimes L}=S^\dagger \), so \(h'=S^\dagger hS\) is positive semidefinite. Since \(S\) and \(S^\dagger \) are invertible with inverses \(\Lambda ^{\otimes L}\) and \((\Lambda ^\dagger )^{\otimes L}\),
where the last equality is (216).
Let \(0{\lt}N\), \(L\le N\), let \(i\in \{ 0,\ldots ,N{-}1\} \), and let \(\tau \in \{ 0,\ldots ,d{-}1\} ^{N}\) be an outside configuration. For every \(\phi \colon \{ 0,\ldots ,d{-}1\} ^{N}\to \mathbb {C}\), the restriction of \(\Lambda ^{\otimes N}\phi \) to the cyclic window \([i,i+L-1]\) with outside configuration \(\tau \) is
where the sum runs over the configurations \(\omega \in \{ 0,\ldots ,d{-}1\} ^{N}\) that agree with \(\tau \) on the window, and \(c_\tau (\omega )=\prod _{k\notin [i,i+L-1]}\Lambda _{\tau _k\omega _k}\) is the product of the entries of \(\Lambda \) over the sites outside the window.
Write \(W=[i,i+L-1]\) for the window. For a configuration \(\omega \in \{ 0,\ldots ,d{-}1\} ^{N}\) let \(\omega _W\in \{ 0,\ldots ,d{-}1\} ^{L}\) be the word read off the window, \((\omega _W)_r=\omega _{i+r}\), and for a window word \(\sigma '\in \{ 0,\ldots ,d{-}1\} ^{L}\) let \(\omega [\sigma ']\in \{ 0,\ldots ,d{-}1\} ^{N}\) be the configuration that reads \(\sigma '\) on the window and agrees with \(\omega \) outside it, so that \((\phi |_{W,\omega })(\sigma ')=\phi (\omega [\sigma '])\), \((\omega [\sigma '])_W=\sigma '\), \(\omega [\sigma '][\omega _W]=\omega \), and \(c_\tau (\omega [\sigma '])=c_\tau (\omega )\). Evaluate both sides of (219) at a window word \(\sigma \). On the left, the product over all sites in (214) splits into the product over the window, where \(\tau [\sigma ]\) reads \(\sigma \), and the product over its complement, where \(\tau [\sigma ]\) agrees with \(\tau \):
On the right, with \([\, \cdot \, ]\) equal to \(1\) when the enclosed condition holds and to \(0\) otherwise,
where (221) reindexes the double sum by the involution
on pairs of a window word and a configuration, and (222) collapses the sum over \(\sigma '\) to the single term \(\sigma '=\tau _W\). The right sides of (220) and (222) agree term by term with \(\rho =\omega \).
Let \(\Lambda \in M_{d}(\mathbb {C})\) be invertible. Then, for all \(L\) and \(N\),
and for \(0{\lt}N\) and \(L\le N\) the kernels of the parent Hamiltonians satisfy \(\ker H_N(\Lambda A)=\Lambda ^{\otimes N}\ker H_N(A)\). This is the matrix-product-state instance, on the periodic chain, of the one-to-one correspondence, through \(R^{\otimes N}\), between the ground states of the parent Hamiltonians of \(|\psi \rangle \) and \(|\psi '\rangle =R^{\otimes N}|\psi \rangle \) for an invertible map \(R\) stated in [ 17 , Section IV.C ] for tensor network states on a general graph. Without invertibility, the inclusion \(\Lambda ^{\otimes N}\mathcal G_{N,L}(A)\subseteq \mathcal G_{N,L}(\Lambda A)\) still holds.
Let \(\phi \in \mathcal G_{N,L}(A)\), so that every window restriction of \(\phi \) lies in \(G_L(A)\). By (219), every window restriction of \(\Lambda ^{\otimes N}\phi \) is \(\Lambda ^{\otimes L}\) applied to a linear combination of window restrictions of \(\phi \), hence lies in \(\Lambda ^{\otimes L}G_L(A)=G_L(\Lambda A)\) by (216). This proves the inclusion \(\Lambda ^{\otimes N}\mathcal G_{N,L}(A)\subseteq \mathcal G_{N,L}(\Lambda A)\) for every \(\Lambda \). Applying it to \(\Lambda ^{-1}\) and \(\Lambda A\) gives \((\Lambda ^{-1})^{\otimes N}\mathcal G_{N,L}(\Lambda A)\subseteq \mathcal G_{N,L}(A)\), and composing with \(\Lambda ^{\otimes N}\) gives the reverse inclusion. The kernel statement follows from Theorem 12.3.17.
Let \(\Lambda \in M_{d}(\mathbb {C})\) be invertible. If \(\dim \mathcal G_{N,L}(A)=1\), then \(\dim \mathcal G_{N,L}(\Lambda A)=1\).
By (224), \(\mathcal G_{N,L}(\Lambda A)\) is the image of \(\mathcal G_{N,L}(A)\) under the linear automorphism \(\Lambda ^{\otimes N}\), which preserves dimension.
11.8 Periodic-boundary comparison for injective tensors
On the periodic ring, the two cyclic supports that cross the periodic-boundary cut expose the same complementary word, and matching their boundary matrices forces \(X\) to commute with every \(A^j\) via the boundary-crossing compatibilities \(C^+_\tau A^j X=Y_\tau A^j\) and \(X A^j C^-_\tau =A^jY_\tau \).
Assume \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\) and \(M\ge L_0\). If the reduced cyclic interval crossing the last site has boundary matrices \(Y_\tau \), then, for every physical letter \(j\), one has
Here \(C^+_\tau \) is the complementary word seen by that cyclic interval.
The trace identity for this cyclic interval is tested against all words of length \(L_0\). Injectivity of \(\Gamma _{L_0}\), which follows from \(L_0\)-block injectivity, turns these test-word identities into (225).
Assume \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\) and \(M\ge L_0\). If the second boundary-crossing reduced cyclic interval has boundary matrices \(Y_\tau \), then \(X A^j C^-_\tau =A^jY_\tau \) for every physical letter \(j\), where \(C^-_\tau \) is the complementary word seen from the second cyclic position.
Factor the cyclic word at the second cyclic position, rotate the trace, and use injectivity of \(\Gamma _{L_0}\) to remove the length-\(L_0\) test word.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Suppose \(\psi \in \mathcal G_{N,L}(A)\), \(N\ge 2\), \(L_0{\lt}L\le N\), and \(\psi =\Gamma _N(X)\) for a boundary matrix \(X\). Then there are two families of boundary matrices \(Y^+_\tau \) and \(Y^-_\tau \), indexed by fixed complement values \(\tau \), such that, for every physical letter \(j\),
Here \(C^+_\tau \) and \(C^-_\tau \) are the complementary words exposed by the two boundary-crossing cyclic intervals, both of reduced length \(L_0+1\).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(L_0\le M\). Suppose \(\psi =\Gamma _{M+1}(X)\), and suppose matrices \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Then, for every physical letter \(j\) and boundary condition \(\tau \),
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(L_0\le M\). Suppose \(\psi =\Gamma _{M+1}(X)\), and suppose matrices \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). For every boundary letter \(\eta \), every physical letter \(j\), and every complementary word \(\mu \),
On a chain of length \(N=M+1\), with \(L_0\le M\), fix a physical letter \(\eta \) used outside the complementary sites and a word \(\mu \) of length \(M+1-(L_0+1)\). Write \(\tau ^+_\eta (\mu )\) for the boundary condition at the support crossing the last site and \(\tau ^-_\eta (\mu )\) for the boundary condition at the second boundary-crossing support. Their exposed complements are both exactly \(\mu \).
Put the same letter \(\eta \) on the remaining sites. Fill the physical sites \(L_0,\ldots ,M-1\) with \(\mu \) for the support crossing the last site, and fill the sites \(1,\ldots ,M-L_0\) with \(\mu \) for the second boundary-crossing support. The two complement-extraction identities follow from the index arithmetic. The same outside-site calculation gives \(\operatorname{Res}^\rho _{M,L_0+1}\psi =\operatorname{Res}^{\tau ^+_\eta (\mu )}_{M,L_0+1}\psi \) for any \(\rho \) with \(\rho _{k+L_0}=\mu _k\), and \(\operatorname{Res}^\rho _{M+1-L_0,L_0+1}\psi =\operatorname{Res}^{\tau ^-_\eta (\mu )}_{M+1-L_0,L_0+1}\psi \) for any \(\rho \) with \(\rho _{k+1}=\mu _k\).
Let \(A\) be \(L_0\)-block-injective, with \(L_0{\gt}0\), and let \(L_0\le M\). Suppose two length-\((L_0+1)\) restrictions of a state \(\psi \) at the same cyclic support are represented by \(\operatorname{Res}^\rho _{i,L_0+1}\psi =\Gamma _{L_0+1}(Y_\rho )\) and \(\operatorname{Res}^\tau _{i,L_0+1}\psi =\Gamma _{L_0+1}(Y_\tau )\). If these two restrictions are equal, then \(Y_\rho A^j=Y_\tau A^j\) for every physical letter \(j\). In particular, the last-site boundary-crossing condition and the second boundary-crossing condition satisfy, respectively,
Consequently, for every word \(\sigma \) of length \(L_0\), both implications remain true after right multiplication by \(A^\sigma \).
Apply the first-letter restriction to both equal length-\((L_0+1)\) restrictions. The two resulting length-\(L_0\) vectors are represented by \(\Gamma _{L_0}(Y_\rho A^j)\) and \(\Gamma _{L_0}(Y_\tau A^j)\), and block injectivity makes \(\Gamma _{L_0}\) injective. The two displayed special cases follow from Lemma 11.8.6. The length-\(L_0\) word forms are obtained by multiplying these equations on the right by \(A^\sigma \).
Let \(A\) be \(L_0\)-block-injective, with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let the matrices \(Y_i(\tau )\) represent its length-\((L_0+1)\) cyclic restrictions, \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a boundary letter \(\eta \) and a complementary word \(\mu \). Let \(\rho ^+\) and \(\rho ^-\) be two boundary conditions satisfying \(\rho ^+_{k+L_0}=\mu _k\) and \(\rho ^-_{k+1}=\mu _k\). If the adjacent-window comparison gives, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\), \(Y_M(\rho ^+)A^jA^\sigma =Y_{M+1-L_0}(\rho ^-)A^jA^\sigma \), then the same product equation holds for \(\tau ^+_\eta (\mu )\) and \(\tau ^-_\eta (\mu )\):
By Lemma 11.8.7,
Multiplying on the right by \(A^\sigma \) and applying the hypothesis gives the displayed product equation.
Let \(A\) be \(L_0\)-block-injective, with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let the matrices \(Y_i(\tau )\) represent its length-\((L_0+1)\) cyclic restrictions, \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a boundary letter \(\eta \) and a complementary word \(\mu \). Suppose that, for each physical letter \(j\) and word \(\sigma \) of length \(L_0\), boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfy, for every complementary position \(k\),
If, for every \(j\) and \(\sigma \),
then, for every \(j\) and \(\sigma \),
Fix \(j\) and \(\sigma \). Lemma 11.8.7 gives
Multiply (228) and (229) on the right by \(A^\sigma \) and use the assumed product equation for the chosen pair \((j,\sigma )\).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\). Suppose that matrices \(Y_r(\rho )\), for \(0\le r\le L_0-1\), represent the length-\((L_0+1)\) cyclic restrictions beginning at the sites \(M+1-L_0+r\): \(\operatorname{Res}^\rho _{M+1-L_0+r,L_0+1}\psi =\Gamma _{L_0+1}(Y_r(\rho ))\). Then
For \(L_0=1\), both products are empty.
Use Theorem 11.5.5.3 for the \(L_0-1\) adjacent restrictions starting at \(i_0=M+1-L_0\). For \(0\le r{\lt}L_0-1\), the two endpoint words are determined by
Substituting these two words in the iterated transport identity gives the displayed product equation.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and suppose matrices \(Y_i(\rho )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\rho _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\rho ))\). Then, for every boundary condition \(\rho \),
For \(L_0=1\), both products are empty. The same equation remains true after multiplying both sides on the right by any matrix \(R\).
Apply Theorem 11.5.5.3 to the \(L_0-1\) restrictions beginning at \(M+1-L_0+r\), with \(Y_r(\rho )=Y_{M+1-L_0+r}(\rho )\). With \(i_0=M+1-L_0\), the site labels satisfy
Hence, for \(r=0,\ldots ,L_0-2\),
Multiplying these \(L_0-1\) equations from left to right gives the displayed product equation. The right-multiplied form follows by multiplying this equality by \(R\).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), and let \(L_0\le M\). Let \(\psi =\Gamma _{M+1}(X)\), and suppose that matrices \(Y_i(\rho )\) represent all length-\((L_0+1)\) cyclic restrictions of \(\psi \). Then, for every boundary condition \(\rho \) and every physical letter \(j\),
For \(\rho =\tau ^-_\eta (\mu )\), this is the transport identity supplied by the adjacent-window argument, with the single factor \(A^j\) following \(Y_{M+1-L_0}(\tau ^-_\eta (\mu ))\). The padded identity, whose corresponding factor is \(A^jA^\sigma \), is supplied separately.
Lemma 11.8.11 gives
The one-sided equation for the window beginning at \(M\) gives \(Y_M(\rho )A^j =A^{\rho _{L_0}}\cdots A^{\rho _{M-1}}A^jX\). Substitution gives the displayed formula.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and suppose matrices \(Y_i(\rho )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\rho _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\rho ))\). Then, for every boundary condition \(\rho \),
Each side has \(M+2-L_0\) one-site factors, with site labels read modulo \(M+1\).
Apply Theorem 11.5.5.3 to the \(M+2-L_0\) restrictions beginning at \(M+r\) modulo \(M+1\). With \(i_0=M\), the site labels satisfy
The final window is \(i_0+M+2-L_0\equiv M+1-L_0\pmod{M+1}\). Substituting these site labels in the iterated product identity gives the displayed equation.
Fix a physical letter \(\eta \) used outside the complementary sites. Suppose the two one-sided boundary matrix families \(Y^+\) and \(Y^-\) have been extracted from the cyclic windows. If, for every word \(\mu \) on the complementary sites, these boundary matrices agree on the boundary conditions built from \(\eta \), \(Y^+_{\tau ^+_\eta (\mu )}=Y^-_{\tau ^-_\eta (\mu )}\), then there is a single family \(Y_\mu \) satisfying
for all letters \(a,b\).
Define \(Y_\mu \) to be the first boundary matrix evaluated on the boundary condition \(\tau ^+_\eta (\mu )\). Lemma 11.8.6 rewrites the first cyclic-window complement as \(\mu \), while the assumed comparison at \(\tau ^-_\eta (\mu )\) rewrites the second boundary matrix to the same \(Y_\mu \).
Suppose a single family of boundary matrices \(Y_\mu \), indexed by words \(\mu \) on the complementary sites, satisfies the two one-sided identities
for every pair of physical letters \(a,b\). Then \(X\) commutes with every word product \(A^aA^\mu A^b\) obtained by adjoining one letter on each side of the word \(A^\mu \).
Let \(A\) be injective with \(D\ge 1\), and let \(\psi =\Gamma _N(X)\) for some boundary matrix \(X\in M_{D}(\mathbb {C})\). If every cyclic restriction of length \(L\) of \(\psi \) lies in \(G_L(A)\), with \(N\ge 2\) and \(1{\lt}L\le N\), then \(X A^j=A^jX\) for every \(j=0,\ldots ,d-1\).
The periodic boundary condition produces matrix identities of the form \(XMW=MY\) for arbitrary test matrices \(M\) and complementary words \(W\). Spanning first over the length-\((L-1)\) tails and then over the complementary words, injectivity forces \((XM-MX)W=0\) for every \(W\), hence \(XM=MX\) for every matrix \(M\). In particular, \(X\) commutes with each letter \(A^j\).
11.9 Periodic unique ground state
For an injective tensor with \(D\ge 1\), \(1{\lt}L\le N\), and \(N\ge 2\) (or an \(L_0\)-block-injective tensor with \(D\ge 1\), \(L_0{\gt}0\), \(L_0{\lt}L\le N\), and \(N\ge 2\)), the periodic matrix product vector \(V^{(N)}(A)=\Gamma _N(\mathbb {1})\) satisfies every cyclic window constraint \(\mathcal G_{N,L}(A)\). Together with the boundary-matrix commutation \(X A^j=A^jX\), this establishes that the periodic ground space is spanned by \(V^{(N)}(A)\).
The subspace spanned by the MPS vector \(V^{(N)}(A)(\sigma )=\operatorname{tr}(A^{\sigma _0}\cdots A^{\sigma _{N-1}})\).
For every chain length \(N\), one has \(V^{(N)}(A)=\Gamma _N(\mathbb {1})\).
\(V^{(N)}(A)\in G_N(A)\) for every \(N\), with boundary matrix \(X=\mathbb {1}\).
For every \(\sigma \), \(V^{(N)}(A)(\sigma )=\operatorname{tr}(A^\sigma )=\operatorname{tr}(A^\sigma \mathbb {1}) =\Gamma _N(\mathbb {1})(\sigma )\).
A finite-dimensional ground space \(S\) is one-dimensional if \(\dim S=1\).
For finite-dimensional \(S\), \(\dim S=1\) if and only if there exists a nonzero \(\psi _0\in S\) such that every \(\psi \in S\) is of the form \(c\, \psi _0\).
Apply the standard finite-dimensional criterion that a nonzero space has dimension one exactly when all of its vectors are proportional to a single nonzero vector.
For \(N{\gt}0\) and \(L\le N\), define the periodic parent-Hamiltonian ground space
Here the condition \(\psi |_{[i,i+L-1]}\in G_L(A)\) means that, for every choice of physical indices outside the window, the restriction of \(\psi \) to that window lies in \(G_L(A)\). The one-index notation \(G_L(A)\) denotes the local ground space \(\mathcal G_L\) of [ 17 , Section IV.C ] . The two-index notation \(\mathcal G_{N,L}(A)\) used here denotes the periodic intersection of those local constraints over all translated length-\(L\) windows on the \(N\)-site ring. When \(N=0\) or \(L{\gt}N\), set \(\mathcal G_{N,L}(A):=\top \) by convention.
If \(0{\lt}N\), \(L\le N\), and \(\psi \in \mathcal G_{N,L}(A)\), then, for every cyclic window and every outside configuration, there is a boundary matrix \(Y\) such that the corresponding restricted \(L\)-site state is \(\Gamma _L(Y)\).
This is the defining cyclic-window condition for membership in \(\mathcal G_{N,L}(A)\), together with the definition of the local ground space as the range of \(\Gamma _L\).
If \(0{\lt}N\) and \(L\le N\), then \(V^{(N)}(A)\in \mathcal G_{N,L}(A)\).
If \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(N\ge L_0+1\), then \(V^{(N)}(A)\ne 0\) in the \(N\)-site space.
Suppose for contradiction that \(V^{(N)}(A)=0\), i.e., \(\operatorname{tr}(A^\sigma )=0\) for every word \(\sigma \) of length \(N\). For any words \(w\) and \(u\) with \(|w|=N-L_0\) and \(|u|=L_0\), we have \(\operatorname{tr}(A^wA^u)=0\). Since \(A\) is \(L_0\)-block-injective, \(\{ A^u:|u|=L_0\} \) spans \(M_{D}(\mathbb {C})\). By nondegeneracy of the trace pairing, \(A^w=0\) for every \(|w|=N-L_0\). Repeatedly strip a length-\(L_0\) suffix: if every product of length \(m+L_0\) vanishes, then the spanning property of the length-\(L_0\) products implies that every product of length \(m\) vanishes. Eventually all products of some length \(r{\lt}L_0\) vanish. If \(r=0\), this gives \(\mathbb {1}=0\). If \(r{\gt}0\), every length-\(L_0\) product has a zero length-\(r\) prefix, so all such products vanish, contradicting that they span \(M_{D}(\mathbb {C})\).
11.10 Injectivity-length reduction for the closure property
Following [ 17 , Section IV.C ] , commutation or annihilation relations for length-\(m\) word products can be reduced to the injectivity length \(L_0\) by word-spanning arguments. Throughout, \(A^w := A^{i_1}\cdots A^{i_L}\) denotes the product along a word \(w = (i_1,\ldots ,i_L)\), \(\Gamma _N\) sends a boundary matrix to its open-chain vector, and \(A\) being \(L_0\)-block-injective means \(\operatorname{span}\{ A^u : |u|=L_0\} = M_{D}(\mathbb {C})\), so a relation valid on every length-\(L_0\) product extends by linearity to all of \(M_{D}(\mathbb {C})\).
Assume \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\). Let \(\{ Z_u\} _{|u|=L_0}\) be a family indexed by words of length \(L_0\). If, for some \(K \ge 0\) and every suffix word \(w\) of length \(K\), there exists \(Y_w \in M_{D}(\mathbb {C})\) such that \(Z_u A^w = A^u Y_w\) for every word \(u\) of length \(L_0\), then there exists \(X \in M_{D}(\mathbb {C})\) with \(Z_u = A^u X\) for every word \(u\) of length \(L_0\).
Induct on \(K\). For \(K = 0\), the empty suffix gives \(Z_u = A^u Y_{\emptyset }\). For the successor step, fix the first physical index \(i\) and apply the induction hypothesis to the shortened suffix. This gives a matrix \(X_i\) satisfying \(Z_u A^i = A^u X_i\) for every word \(u\) of length \(L_0\). Extending this letter equation to all length-\(L_0\) words and decomposing \(\mathbb {1}\) in the spanning family \(\{ A^u : |u| = L_0\} \) yields a common factor \(X\).
Let \(A\) be a tensor whose length-\(K\) word products span \(M_{D}(\mathbb {C})\). Let \(\{ F_b\} _{b\in B}\) and \(\{ Z_b\} _{b\in B}\) be two families of matrices. If, for every length-\(K\) word \(w\), there is a matrix \(Y_w\) such that \(Z_b A^w = F_bY_w\) for every \(b\in B\), then there is a single matrix \(Y\) such that \(Z_b = F_bY\) for every \(b\in B\).
Suppose \(Z_bM_1 = F_bY_1\) and \(Z_bM_2 = F_bY_2\) for every \(b\in B\). Then \(Z_b(M_1+M_2) = Z_bM_1+Z_bM_2 = F_b(Y_1+Y_2)\). For a scalar \(c\), one similarly has \(Z_b(cM_1) = cZ_bM_1 = F_b(cY_1)\). Thus the identity holds for every matrix in the span of the length-\(K\) word products. Since this span is all of \(M_{D}(\mathbb {C})\), apply it to the identity matrix.
If \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\), \(m \ge L_0\), and \(X \in M_{D}(\mathbb {C})\) commutes with every length-\(m\) product \(A^\omega \), then \(X\) already commutes with every length-\(L_0\) product \(A^u\).
Write each length-\(m\) word as a concatenation \(uw\) with \(|u| = L_0\) and \(|w| = m - L_0\). The hypothesis gives \(X A^u A^w = A^u A^w X = A^u(A^w X)\). Thus the family \(Z_u := X A^u\) satisfies the hypothesis of Theorem 11.10.1. One obtains \(X A^u = A^u R\) for all \(|u| = L_0\). Decomposing \(\mathbb {1}\) in the span of the length-\(L_0\) words shows \(R = X\), hence \(X A^u = A^u X\).
If \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\), \(m \ge L_0\), and \(X \in M_{D}(\mathbb {C})\) commutes with every length-\(m\) word product, then \(X\) commutes with every matrix in \(M_{D}(\mathbb {C})\).
By Theorem 11.10.3, \(X\) commutes with every length-\(L_0\) word product. These products span \(M_{D}(\mathbb {C})\) by block injectivity, so linearity gives \(XM = MX\) for every \(M \in M_{D}(\mathbb {C})\).
If a boundary matrix \(X\) commutes with every word product of length \(m\), then it commutes with every word product whose length is a multiple \(q\, m\).
Proceed by induction on \(q\). For \(q=0\), the claim is trivial. For the inductive step, let a word of length \((q+1)m\) be given and write it as a length-\(m\) prefix followed by a length-\(q\, m\) suffix. The prefix commutes with \(X\) by hypothesis, and the suffix commutes by the induction hypothesis.
If \(A\) is \(L_0\)-block-injective, \(q \ge 1\), and \(k \le qL_0\), then the two one-sided annihilation statements are
The second implication is the left-handed form.
The hypothesis \(\operatorname{span}\{ A^u : |u|=L_0\} =M_{D}(\mathbb {C})\) implies the same spanning statement at every positive multiple \(qL_0\). Each length-\(qL_0\) word factors as a length-\(k\) prefix followed by padding, so \(Z\) annihilates all generators of the full span and hence annihilates the identity. For the left-handed form, factor each length-\(qL_0\) word as a prefix followed by a length-\(k\) suffix and use the same spanning argument.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). If two boundary matrices have the same product with every one-site tensor on one fixed side, then the two boundary matrices are equal.
If \(Y_1 A^j = Y_2 A^j\) for every physical index \(j\), then induction on the word gives \(Y_1 A^w = Y_2 A^w\) for every nonempty word \(w\). Since \(A\) is \(L_0\)-block-injective, \(\operatorname{span}\{ A^w : |w|=L_0\} =M_{D}(\mathbb {C})\). Thus the left multiplication maps \(M\mapsto Y_1 M\) and \(M\mapsto Y_2 M\) are equal on \(M_{D}(\mathbb {C})\), and evaluating them on the identity gives \(Y_1=Y_2\). The other one-sided statement follows by the same argument, with right multiplication in place of left multiplication.
Under the hypotheses of Theorem 11.10.4, one has \(X A^i = A^i X\) for every physical index \(i\).
Apply Theorem 11.10.4 with \(M = A^i\).
Let \(A\) be injective after blocking \(L_0{\gt}0\) sites, and let \(m \ge L_0\). If a boundary matrix \(X\) commutes with every length-\(m\) word product, then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).
Theorem 11.10.4 makes \(X\) commute with the whole matrix algebra. The center of \(M_{D}(\mathbb {C})\) consists of scalar matrices, so \(X=c\mathbb {1}\), and therefore \(\Gamma _N(X)=cV^{(N)}(A)\).
Let \(A\) be injective after blocking \(L_0{\gt}0\) sites. If \(X\) commutes with every word product of some positive length \(m\), then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).
Let \(A\) be injective after blocking \(L_0{\gt}0\) sites. Suppose there are matrices \(Y_\mu \) such that
for every pair of physical letters \(a,b\). Then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).
Lemma 11.8.15 gives commutation with all words of length \(|\mu |+2\), a positive length. The positive-length containment theorem then applies.
Let \(A\) be injective after blocking \(L_0{\gt}0\) sites, and fix a physical letter \(\eta \) used outside the complementary sites. Suppose that, for every boundary condition \(\tau \) and every physical letter \(j\),
where \(C^+_\tau \) and \(C^-_\tau \) are the complementary words exposed by the two boundary-crossing cyclic windows. Suppose also that the two boundary matrices satisfy \(Y^+_{\tau ^+_\eta (\mu )} = Y^-_{\tau ^-_\eta (\mu )}\), for every word \(\mu \) on the complementary sites. Then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).
Let \(A\) be injective after blocking \(L_0{\gt}0\) sites and let \(D \ge 1\). Assume \(N \ge 2\) and \(L_0 {\lt} L \le N\). Suppose that, for every physical letter \(\eta \) used in the two boundary conditions and every boundary matrix obtained from an \(N\)-site chain ground state, the identities
hold for every physical letter \(j\) and every complementary word \(\mu \), and are supplemented, for every \(\mu \), by \(Y^+_{\tau ^+_\eta (\mu )} = Y^-_{\tau ^-_\eta (\mu )}\). Then \(\mathcal G_{N,L}(A)\subseteq \operatorname{span}\{ V^{(N)}(A)\} \).
The cyclic-to-open-chain reduction writes any chain ground state as \(\Gamma _N(X)\). The two boundary-crossing cyclic windows give
Theorem 11.10.12 finishes.
11.11 Periodic-boundary comparison for normal tensors
For normal tensors, the periodic-boundary argument closes open intervals into periodic chains using boundary-crossing restrictions [ 17 , Section IV.C ] . For a vector \(\psi =\Gamma _{M+1}(X)\), the closure property requires that the two boundary-crossing restrictions \(\operatorname{Res}^{\tau ^+_\eta (\mu )}_{M,L_0+1}(\psi )\) and \(\operatorname{Res}^{\tau ^-_\eta (\mu )}_{M+1-L_0,L_0+1}(\psi )\) coincide. The boundary conditions \(\tau ^\pm _\eta (\mu )\) parametrize these restrictions, and the matrices \(Y_i(\tau )\) satisfy \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Once the one-sided boundary products are known, equality of the restrictions follows from the boundary-matrix commutation relation \(XA^j=A^jX\) for every physical letter \(j\). The block-window identity \(X A^\alpha A^\nu =A^\alpha Y_\nu \) is derived from the cyclic-window constraints crossing the periodic cut.
Let \(A\) be an MPS tensor and let \(L_0,K\ge 0\). Let \(X\) be a matrix, and let \(Y_{c_b}\) be a matrix for each iterated block index \(c_b\) of the complement. Suppose that, for every block letter \(b\) of the alphabet \(\{ 0,\ldots ,d{-}1\} ^{L_0}\) and every iterated block index \(c_b\) of the complement, with \(w(b)\) the length-\(L_0\) word of \(b\) and \(\widetilde{w(c_b)}\) the length-\(L_0K\) complement word obtained by concatenating the blocks of \(c_b\), \(X A^{w(b)} A^{\widetilde{w(c_b)}} = A^{w(b)} Y_{c_b}\). Then there is a family \(Y'_c\), indexed by the length-\(L_0K\) words \(c\), with \(X A^s A^c = A^s Y'_c\) for every length-\(L_0\) word \(s\) and every length-\(L_0K\) word \(c\).
Each length-\(L_0\) word \(s\) is the word \(w(b)\) of its block letter \(b\), and each length-\(L_0K\) word \(c\) regroups into \(K\) blocks, recovering it as the concatenated complement word \(\widetilde{w(c_b)}\) of an iterated block index \(c_b\); both identifications are bijective. Substituting these representations into the hypothesis and setting \(Y'_c=Y_{c_b}\) gives the asserted equation for every \(s\) and \(c\).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Let \(X\) be a boundary matrix, and let \(Y_v\) be a matrix for each complementary word \(v\) of length \(K\). Suppose that, for every word \(u\) of length \(L_0\) and every word \(v\) of length \(K\), \(X A^u A^v = A^u Y_v\). Then, for every physical letter \(j\), \(X A^j = A^j X\). This establishes the boundary-matrix commutation step of [ 17 , Section IV.C, lines 2049–2090 ] .
By Lemma 6.1.1.23, \(\operatorname{span}\{ A^u : |u|=L_0\} = M_{D}(\mathbb {C})\). Hence, for every \(M\in M_{D}(\mathbb {C})\) and every complementary word \(v\) of length \(K\), the displayed hypothesis extends by linearity in the length-\(L_0\) block to \(XMA^v=MY_v\). Taking \(M=\mathbb {1}\) gives
Taking \(M=A^j\) and substituting (232) gives \((X A^j-A^jX)A^v=0\) for every word \(v\) of length \(K\). Since \(K\le (K+1)L_0\) and \(K+1\ge 1\), Lemma 11.10.6 gives \(X A^j-A^jX=0\).
Let \(A\) be a tensor with virtual dimension at least one, and let \(L_0{\gt}0\) and \(L_0{\lt}M\). Let \(\psi =\Gamma _{M+1}(X)\) and assume that every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\). Then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every physical letter \(j\) and every word \(\alpha \) of length \(L_0\),
Let \(Y_\nu \) be the matrix representing the cyclic restriction whose window begins at the last site. The cyclic word is \(\alpha \, \nu \, j\) after deleting the first letter from the window and reading the complement, so the ground-space representation and cyclicity of the trace give
Let \(A\) be a tensor with virtual dimension at least one whose single-site matrices span the full matrix algebra. Let \(L_0{\gt}0\) and \(L_0{\lt}M\), and let \(\psi =\Gamma _{M+1}(X)\). If every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\), then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every word \(\alpha \) of length \(L_0\), \(X A^\alpha A^\nu = A^\alpha Y_\nu \).
For every physical letter \(j\), Lemma 11.11.3 gives
Put \(Z=X A^\alpha A^\nu -A^\alpha Y_\nu \). For every physical letter \(j\), (233) gives \(\operatorname{tr}(A^jZ)=0\). Since the matrices \(A^j\) span the full matrix algebra, the trace pairing gives \((\forall j,\ \operatorname{tr}(A^jZ)=0) \quad \Longrightarrow \quad Z=0\). Hence \(X A^\alpha A^\nu =A^\alpha Y_\nu \).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) and assume that every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\). Then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every word \(\alpha \) of length \(L_0\) and every word \(\beta \) of length \(L_0\),
Choose representation matrices for every cyclic restriction of length \(L_0+1\). Fix \(\alpha \) and \(\nu \). The boundary-crossing equations move the matrix \(X\) through the first boundary letter. In local boundary notation, this has the form
The adjacent boundary-window product then transports this comparison through the remaining \(L_0-1\) boundary letters:
Finally, the outside-label uniqueness lemma identifies the final boundary matrix with the matrix \(Y_\nu \) attached to the complementary word \(\nu \). Thus \(X A^\alpha A^\nu =A^\alpha Y_\nu \). Multiplying on the left by \(A^\beta \) and taking the trace gives the asserted identity for every length-\(L_0\) word \(\beta \).
Let \(A\) be \(L_0\)-block-injective. Let \(X\) be a boundary matrix, and let \(Y_\nu \) be a matrix for each complementary word \(\nu \) of length \(K\). Suppose that, for every word \(\alpha \) of length \(L_0\), every word \(\beta \) of length \(L_0\), and every complementary word \(\nu \),
Then, for every word \(\alpha \) of length \(L_0\) and every complementary word \(\nu \), \(X A^\alpha A^\nu = A^\alpha Y_\nu \).
The trace identities say exactly that, for each fixed pair \((\alpha ,\nu )\),
Since \(A\) is \(L_0\)-block-injective, \(\Gamma _{L_0}\) is injective by Theorem 11.5.3.3. Hence \(X A^\alpha A^\nu =A^\alpha Y_\nu \).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) and assume that every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\). Then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every word \(\alpha \) of length \(L_0\), \(X A^\alpha A^\nu = A^\alpha Y_\nu \).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Let \(X\) be a boundary matrix, let \(W_\eta \) and \(V_\eta \) be matrix families indexed by a physical letter \(\eta \), and let \(\mu \) be a word. Suppose that, for every physical letter \(j\), \(X A^j = A^jX\). Suppose also that, for every \(\eta \) and every \(j\),
Then, for every \(\eta \) and every \(j\), \(W_\eta A^j = V_\eta A^j\).
The commutation relation \(X A^j=A^jX\) extends by induction to \(X A^w=A^wX\) for every word \(w\). Hence the second one-sided equation gives, for every \(k\), \(A^kV_\eta = X A^kA^\mu = A^kX A^\mu = A^kA^\mu X\). Lemma 11.10.7 gives \(V_\eta =A^\mu X\). The first one-sided equation then gives \(W_\eta A^j = A^\mu A^jX = A^\mu X A^j = V_\eta A^j\).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Fix two matrix families \(Y^+\) and \(Y^-\) and the two reindexed boundary conditions \(\tau ^+_\eta (\mu )\) and \(\tau ^-_\eta (\mu )\). If, for every physical letter \(j\), \(Y^+_{\tau ^+_\eta (\mu )} A^j = Y^-_{\tau ^-_\eta (\mu )} A^j\), then \(Y^+_{\tau ^+_\eta (\mu )} = Y^-_{\tau ^-_\eta (\mu )}\).
Apply Lemma 11.10.7 to the two matrices \(Y^+_{\tau ^+_\eta (\mu )}\) and \(Y^-_{\tau ^-_\eta (\mu )}\).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(L_0\le M\). If an outside configuration \(\rho \) has the same word \(\mu \) as \(\tau ^+_\eta (\mu )\) on the sites outside the last-site cyclic window, then the two matrices representing that restriction agree: \(Y_\rho = Y_{\tau ^+_\eta (\mu )}\). The same assertion holds for the second boundary-crossing window: if \(\rho \) has the same outside word as \(\tau ^-_\eta (\mu )\), then \(Y_\rho = Y_{\tau ^-_\eta (\mu )}\). This is the boundary-matrix independence step used to choose one representative outside configuration for each outside word in the closure-property matrix comparison.
Equality of the cyclic restrictions gives, for every physical letter \(j\), \(Y_\rho A^j = Y_{\tau ^\pm _\eta (\mu )}A^j\). Lemma 11.10.7 identifies the two matrices from these one-site products.
Let \(R_j\) be restriction to first letter \(j\):
For \(\phi ,\psi \in (\mathbb {C}^d)^{\otimes (L+1)}\),
At \((\sigma _0,\sigma _1,\ldots ,\sigma _L)\), the required equality is
Let \(L_0{\gt}0\) and \(L_0\le M\), and set
With \(R_j\) as in Lemma 11.11.11,
Apply Lemma 11.11.11 to the two length-\((L_0+1)\) restrictions, using the displayed family of first-letter equalities.
Let \(A\) be an MPS tensor, let \(L_0{\gt}0\), and let \(L_0\le M\). Suppose the two boundary-crossing restrictions of length \(L_0+1\) are represented by boundary matrices \(Y^+_{\eta ,\mu }\) and \(Y^-_{\eta ,\mu }\):
If, for every physical letter \(j\), \(Y^+_{\eta ,\mu }A^j = Y^-_{\eta ,\mu }A^j\), then \(B^+_{\eta ,\mu }(\psi ) = B^-_{\eta ,\mu }(\psi )\).
Fix \(j\) and take the first-letter restriction of the two displayed length-\((L_0+1)\) restrictions. These are \(\Gamma _{L_0}(Y^+_{\eta ,\mu }A^j)\) and \(\Gamma _{L_0}(Y^-_{\eta ,\mu }A^j)\). The one-site product equality gives, for every \(j\),
Lemma 11.11.11 identifies the two original restrictions.
Let \(L_0{\gt}0\) and \(L_0\le M\). Let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a complementary word \(\mu \). If, for every boundary letter \(\eta \) and physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\), then, for every \(\eta \),
Apply Lemma 11.11.13 for each boundary letter \(\eta \), using the corresponding first-letter product equations.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), let \(L_0{\lt}M\), and let
For every boundary letter \(\eta \) and complementary word \(\mu \), the two restrictions crossing the periodic cut satisfy
The cyclic-window representation lemma gives matrices \(Y_M(\tau ^+_\eta (\mu ))\) and \(Y_{M+1-L_0}(\tau ^-_\eta (\mu ))\) such that
Theorem 11.11.1.8 gives, for every \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\). Lemma 11.11.13 identifies the two length-\((L_0+1)\) restrictions.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Suppose that, for every outside letter \(\eta \),
Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) such that, for every complementary position \(k\),
and, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\),
For the product indexed by \(j\) and \(\sigma \), specialize (234) at the outside letter \(\eta =j\). Take \(\rho ^+_{j,\sigma }=\tau ^+_j(\mu )\) and \(\rho ^-_{j,\sigma }=\tau ^-_j(\mu )\). The complement equations are the two identities of Lemma 11.8.6. Fix \(j\). Applying the first-letter restriction to the displayed equality gives
where the two sides are identified by Theorem 11.5.5.3. By Theorem 11.5.3.3, \(Y_M(\tau ^+_j(\mu ))A^j = Y_{M+1-L_0}(\tau ^-_j(\mu ))A^j\). Right multiplication by \(A^\sigma \) gives the asserted product equation.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a complementary word \(\mu \) of length \(M+1-(L_0+1)\). Suppose that, for every boundary letter \(\eta \) and every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\). Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) such that, for every complementary position \(k\),
and, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\),
For each boundary letter \(\eta \), Lemma 11.11.13 gives
Applying these equalities, Lemma 11.11.16 gives the displayed boundary conditions and product equation.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Fix a complementary word \(\mu \), a matrix \(X\), and a family of matrices \(Y_i(\tau )\). Suppose that, for every boundary letter \(\eta \), every physical letter \(j\), and every word \(\sigma \) of length \(L_0\),
Then, for every \(\eta \) and \(j\), \(Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j = A^\mu A^jX\).
Fix \(\eta \) and \(j\), and set \(Z_{\eta ,j} = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j-A^\mu A^jX\). The hypothesis gives \(Z_{\eta ,j}A^\sigma =0\) for every \(\sigma \) with \(|\sigma |=L_0\). Since \(A\) is \(L_0\)-block-injective, the products \(A^\sigma \) with \(|\sigma |=L_0\) span the full matrix algebra. Applying Lemma 11.10.6 gives \(Z_{\eta ,j}=0\).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Fix a complementary word \(\mu \), a matrix \(X\), and a family of matrices \(Y_i(\tau )\). Suppose that, for every boundary letter \(\eta \), every physical letter \(j\), and every pair of words \(\sigma ,\alpha \) of length \(L_0\),
Then, for every \(\eta \), \(j\), and \(\sigma \),
Fix \(\eta \), \(j\), and \(\sigma \), and set
The hypothesis says that \(A^\alpha Z_{\eta ,j,\sigma }=0\) for every word \(\alpha \) of length \(L_0\). The left-handed form of Lemma 11.10.6 gives \(Z_{\eta ,j,\sigma }=0\).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a complementary word \(\mu \) and a matrix \(X\). Suppose that, for every boundary letter \(\eta \) and every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = A^\mu A^jX\). Suppose also that, for every \(\eta \), \(j\), and every word \(\sigma \) of length \(L_0\),
Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) such that, for every complementary position \(k\),
and, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\),
Lemma 11.11.18 gives, for every \(\eta \) and \(j\), \(Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j = A^\mu A^jX\). Combining this equation with \(Y_M(\tau ^+_\eta (\mu ))A^j=A^\mu A^jX\) gives \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\). Lemma 11.11.17 then gives the displayed boundary conditions and product equation.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a complementary word \(\mu \) and a matrix \(X\). Suppose that, for every boundary letter \(\eta \) and every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j=A^\mu A^jX\). Suppose also that, for every \(\eta \), \(j\), and every pair of words \(\sigma ,\alpha \) of length \(L_0\),
Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfying the complementary-word equations
and the product equation
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a complementary word \(\mu \). Suppose that, for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\sigma ,\alpha \),
Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfying the complementary-word equations
and the product equation
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0\le M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a complementary word \(\mu \). Suppose that, for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),
Then, for every boundary letter \(\eta \),
The assumed equations and Lemma 11.11.19 give, for every \(\eta \), \(j\), and \(\sigma \),
Lemma 11.11.18 removes the right word: \(Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j = A^\mu A^jX\). The equation for the boundary-crossing window beginning at \(M\), from Lemma 11.8.5, gives \(Y_M(\tau ^+_\eta (\mu ))A^j = A^\mu A^jX\). Hence the first-letter products agree, and Lemma 11.11.14 identifies the two restrictions.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation. Assume that, for all \(i\) and \(\tau \), \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )\in G_{L_0+1}(A)\). In local coordinate notation, fix a nonempty word \(\mu \) on the sites outside the two cyclic windows. For every outside letter \(\eta \), choose the two outside configurations so that, for each index \(k\) of this outside word,
and assign the letter \(\eta \) to the remaining sites, following the coordinate parametrization of the periodic-boundary comparison in [ 17 , Section IV.C, lines 2078–2079 ] . The corresponding boundary-crossing restriction equality is
Choose matrices \(Y_i(\tau )\) representing the length-\((L_0+1)\) restrictions. The one-sided boundary-product lemma gives, for every outside letter \(\eta \) and physical letter \(j\),
Lemma 11.11.7 gives matrices \(Y_\nu \) satisfying, for every word \(\alpha \) of length \(L_0\) and every complementary word \(\nu \), \(XA^\alpha A^\nu = A^\alpha Y_\nu \). Lemma 11.11.2 then gives, for every physical letter \(k\), the commutation identity \(XA^k=A^kX\). Lemma 11.11.8 gives \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\). Lemma 11.11.14 then identifies the two boundary-crossing restrictions, giving the periodic-boundary comparison of [ 17 , lines 2078–2079 ] .
11.11.1 Reverse comparison for boundary-crossing restrictions
Conversely, equality of the two cyclic restrictions yields the first-letter product equation \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\) and the left-multiplied boundary comparison.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(L_0\le M\). Let \(\psi \) be a vector on \(M+1\) sites, and suppose the two length-\((L_0+1)\) boundary restrictions are represented by \(\Gamma _{L_0+1}(Y_M(\tau ^+_\eta (\mu )))\) and \(\Gamma _{L_0+1}(Y_{M+1-L_0}(\tau ^-_\eta (\mu )))\). If these two restrictions are equal, then for every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\).
Restricting both sides to the first physical letter \(j\) gives
Since \(A\) is \(L_0\)-block-injective, \(\Gamma _{L_0}\) is injective, and the displayed product equality follows.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(L_0\le M\). Let \(\psi \) be a vector on \(M+1\) sites, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Assume that for every boundary letter \(\eta \),
Then for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),
Lemma 11.11.1.1 gives \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\). Multiplying on the left by \(A^\alpha \) and on the right by \(A^\sigma \) gives the desired comparison.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a nonempty complementary word \(\mu \). Then for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),
The matrices \(Y_i(\tau )\) give the local ground-space representations needed in Lemma 11.11.1.2. Once that boundary-restriction equality is proved, it gives
Applying the left-multiplied comparison for equal boundary restrictions gives
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a nonempty complementary word \(\mu \). Then for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),
Once the comparison of the two cyclic restrictions is available, it gives
The one-sided equation for the boundary-crossing window beginning at \(M\) gives \(Y_M(\tau ^+_\eta (\mu ))A^j = A^\mu A^jX\). Substitution gives the displayed identity.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), let \(L_0{\lt}M\), and let \(\psi \in \mathcal G_{M+1,L_0+1}(A)\) with \(\psi =\Gamma _{M+1}(X)\). Let \(R_j\) denote restriction to the first physical letter \(j\). For every outside letter \(\eta \), complementary word \(\mu \), and physical letter \(j\), one has
This provides the single-letter product form of the periodic-boundary comparison in [ 17 , Section IV.C, lines 2078–2079 ] .
Let \(Y_i(\tau )\) be the representation matrices supplied by Lemma 11.9.7. By Theorem 11.11.1.8, \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\). Fixing the first physical letter in the two length-\((L_0+1)\) windows gives
The two restrictions are therefore equal.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Let \(\psi \) be a state on the chain of length \(M+1\) with open-chain representation \(\psi =\Gamma _{M+1}(X)\). Let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi ) = \Gamma _{L_0+1}(Y_i(\tau ))\). If \(L_0{\lt}M\), then for every nonempty complementary word \(\mu \) there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\), depending on the physical letter \(j\) and the word \(\sigma \) of length \(L_0\), such that, for every complementary position \(k\),
and, for every \(j\) and \(\sigma \),
Theorem 11.11.1.4 gives, for every boundary letter \(\eta \), physical letter \(j\), and words \(\alpha ,\sigma \) of length \(L_0\),
Theorem 11.11.22 applies this left-multiplied comparison together with the one-sided boundary products and gives the displayed boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\), including
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Let \(\psi \in \mathcal G_{M+1,L_0+1}(A)\) have an open-chain representation \(\psi =\Gamma _{M+1}(X)\). Let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi ) = \Gamma _{L_0+1}(Y_i(\tau ))\). If \(L_0{\lt}M\), then for every boundary letter \(\eta \), complementary word \(\mu \), physical letter \(j\), and word \(\sigma \) of length \(L_0\),
Theorem 11.11.1.6 gives boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfying
and
Theorem 11.8.9 replaces these auxiliary conditions by the displayed boundary conditions:
which is the product equality needed for the periodic-boundary comparison.
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Let \(\psi \in \mathcal G_{M+1,L_0+1}(A)\) have an open-chain representation \(\psi =\Gamma _{M+1}(X)\). Let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi ) = \Gamma _{L_0+1}(Y_i(\tau ))\). If \(L_0{\lt}M\), then for every letter \(\eta \) used in the two boundary conditions, every word \(\mu \) on the complementary sites, and every physical letter \(j\),
This is the one-site product equality used to compare the two boundary matrices in the periodic-boundary comparison.
Fix \(j\) and set
Theorem 11.11.1.7 gives, after subtracting the two sides, \(Z_jA^\sigma =0\) for every word \(\sigma \) of length \(L_0\). Since \(A\) is \(L_0\)-block-injective, \(\operatorname{span}\{ A^\sigma :|\sigma |=L_0\} = M_D(\mathbb {C})\). Applying Lemma 11.10.6 gives \(Z_j=0\), hence
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), let \(N\ge 2\), \(L_0+1{\lt}N\), and \(L_0{\lt}L\le N\), and let \(\psi \in \mathcal G_{N,L}(A)\) have an open-chain representation \(\psi =\Gamma _N(X)\). Suppose \(Y^{+}\) and \(Y^{-}\) are two families of matrices obtained from the two boundary-crossing cyclic-window constraints, and that they satisfy, for every physical letter \(j\) and every boundary condition \(\tau \),
For every physical letter \(\eta \in \{ 0,\ldots ,d{-}1\} \) and every word \(\mu \) on the complementary sites, define two boundary conditions by
Then the two families agree on these two boundary conditions: \(Y^{+}_{\tau ^{+}_{\eta }(\mu )} = Y^{-}_{\tau ^{-}_{\eta }(\mu )}\). This is the boundary-matrix comparison required for equality of the two restrictions crossing the periodic cut.
The reduced cyclic-window compatibilities at the boundary give, for every physical letter \(j\),
By Lemma 11.11.9, it is enough to prove, for every \(j\),
Write \(N=M+1\). The two boundary-crossing supports start at \(M\) and \(M+1-L_0\). Let \(Y_i(\tau )\) denote the cyclic-window representation matrix given by Lemma 11.9.7 at the window beginning at \(i\). The first step identifies the two abstract boundary-condition families with the two boundary-restriction matrices:
Together with (239), Theorem 11.11.1.8 gives, for every physical letter \(j\),
Lemma 11.11.9 then gives the desired equality of boundary matrices.
Let \(A\) be any tensor. If
then \(\Gamma _{L_0}(XA^j)=\Gamma _{L_0}(A^jY)\) for every physical letter \(j\).
Evaluate the translated equality on a word \(\beta j\), where \(|\beta |=L_0\). Cyclic translation exchanges the two parts of the word, and cyclicity of the trace gives
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). If
then \(XA^j=A^jY\) for every physical letter \(j\).
Theorem 11.11.1.10 gives
and injectivity of \(\Gamma _{L_0}\) gives \(XA^j=A^jY\).
Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), and let \(\psi \in \mathcal G_{L_0+1,L_0+1}(A)\) satisfy \(\psi =\Gamma _{L_0+1}(X)\). Then \(XA^j=A^jX\) for every physical letter \(j\).
Write \(s\) for the cut shift by \(L_0\) and \(Y_i\) for the boundary matrix of the translated state at cut \(i\). Translation invariance of the chain space, the cyclic open-chain reduction, and the composition law supply compatible matrices at successive cuts. Theorem 11.11.1.11 gives the single-letter comparison, and induction on the word \(w\) gives
Injectivity of the propagated length-\((L_0+1)\) boundary map gives \(Y_0=X\). Since \((L_0+1)s=0\), every full-ring word \(\omega \) satisfies \(XA^\omega =A^\omega X\). Lemma 11.10.8 then gives \(XA^j=A^jX\) for every letter \(j\).
If \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), then
Let \(\psi \in \mathcal G_{L_0+1,L_0+1}(A)\). The cyclic open-chain reduction gives \(\psi =\Gamma _{L_0+1}(X)\). Theorem 11.11.1.12 shows that \(X\) commutes with every letter, and hence with every word of length \(L_0\). The long-word commutation theorem then places \(\psi \) in \(\operatorname{span}\{ V^{(L_0+1)}(A)\} \).
11.11.2 Normal unique-ground-state consequences
For injective tensors with \(N\ge 2\) and \(1{\lt}L\le N\), the chain ground space equals \(\operatorname{span}\{ V^{(N)}(A)\} \), ensuring a unique ground state for the parent Hamiltonian. For normal tensors, uniqueness holds for \(N\ge 2\), \(L_0+1\le N\), and \(L_0{\lt}L\le N\) by combining the boundary-crossing comparison with the open-chain range reduction. In the minimal-ring case \(N=L_0+1\), the full-ring change-of-cut theorem replaces the complementary-word comparison.
If \(A\) is injective, \(D \ge 1\), \(N \ge 2\), and \(1 {\lt} L \le N\), then \(\mathcal G_{N,L}(A) = \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).
Let \(\psi \in \mathcal G_{N,L}(A)\). The non-wrapping window conditions and Lemma 11.5.5.4 imply \(\psi \in G_{N}(A)\), so \(\psi = \Gamma _{N}(X)\) for some boundary matrix \(X\). The periodic boundary condition now applies Theorem 11.8.16 and forces \(X\) to commute with every letter \(A^{j}\). Since \(A\) is injective, the letters generate the full matrix algebra, hence \(X\) is scalar. Therefore \(\psi \) is proportional to \(V^{(N)}(A)\), while the reverse inclusion is Lemma 11.9.8.
If \(A\) is normal, \(D \ge 1\), and injective after blocking \(L_0{\gt}0\) sites, with \(N \ge 2\), \(L_0+1\le N\), and \(L_0 {\lt} L \le N\), then \(\mathcal G_{N,L}(A) \subseteq \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).
If \(N=L_0+1\), range antitonicity gives
and Theorem 11.11.1.13 applies to the minimal ring. If \(N{\gt}L_0+1\), the open-chain containment gives \(\psi =\Gamma _N(X)\). The two boundary-crossing cyclic-window constraints give
The boundary-crossing comparison is \(Y^+_{\tau ^+_\eta (\mu )}=Y^-_{\tau ^-_\eta (\mu )}\). These equations imply \(\psi \in \operatorname{span}\{ V^{(N)}(A)\} \).
If \(A\) is normal, \(D \ge 1\), and injective after blocking \(L_0{\gt}0\) sites, with \(N \ge 2\), \(L_0+1\le N\), and \(L_0 {\lt} L \le N\), then \(\mathcal G_{N,L}(A) = \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).
If \(D \ge 1\), \(A\) is injective after blocking \(L_0{\gt}0\) sites, and \(L_0 {\lt} L \le N\), then the parent Hamiltonian of interaction range \(L\) has a unique periodic ground state. These inequalities imply \(N\ge 2\) and \(N\ge L_0+1\), extending the finite-size range \(N\ge 2L_0\) of [ 228 , Theorem 10 ] .
Block injectivity at the positive length \(L_0\) makes \(A\) normal. Hence Theorem 11.11.2.3 gives
The block-injectivity nonvanishing theorem gives
Therefore
For an injective tensor \(A\) with \(D \ge 1\), if \(N \ge 2\) and \(1 {\lt} L \le N\), then the periodic chain ground space is one-dimensional.
By Theorem 11.11.2.1, the ground space is the line \(\operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \). It remains to show that the MPV is nonzero. If \(V^{(N)}(A)=0\), then Lemma 11.9.2 gives \(\Gamma _{N}(\mathbb {1})=\Gamma _{N}(0)\). Since \(A\) is injective and \(N {\gt} 0\), Theorem 11.5.3.1 forces \(\mathbb {1}= 0\), a contradiction. Therefore the spanning line is one-dimensional.
If \(A\) becomes injective after blocking \(L_0 {\gt} 0\) sites and \(D \ge 1\), the parent Hamiltonian with interaction range \(2L_0\) has a unique ground state on every periodic chain with \(N \ge 2L_0\). In particular, this includes \(L_0=1\) and \(N=2\).
Since \(L_0{\gt}0\), one has \(L_0{\lt}2L_0\). The result is therefore the case \(L=2L_0\) of Theorem 11.11.2.4.
If \(A\) is normal, becomes injective after blocking \(L_0 {\gt} 0\) sites, and \(D \ge 1\), then the parent Hamiltonian with interaction range \(L_0 + 1\) has a unique ground state on every periodic chain with \(L_0+1\le N\).
The inequalities \(L_0{\lt}L_0+1\leq N\) put this statement in the range of Theorem 11.11.2.4.
11.12 Half-chain Schmidt spectrum and the dual fixed point
On a ring of \(2L\) sites split into two halves of \(L\) sites each, the matrix product vector splits at the two boundaries into a pair of half-chain families whose overlaps are matrix elements of the \(L\)-fold transfer map [ 228 , Theorem 6 ] . The unnormalized half-chain reduced density matrix factors through the half-chain Gram matrix. Away from zero, its spectrum matches that of a boundary comparison matrix of size \(D^2\), which, under the spectral gap condition \(\rho _{\mathrm{spec}}(\mathcal{E}_A - P_\Lambda ) {\lt} 1\), converges entrywise, geometrically in \(L\), to the diagonal matrix with entries \(\lambda _\alpha \lambda _\beta /\operatorname{tr}(\Lambda )^2\) determined by the fixed point \(\Lambda \).
The source states its canonical form with a trace-preserving dual map and a positive diagonal fixed point \(\Lambda \) of that dual. Here the trace-preserving direction is placed on the transfer map itself and \(\Lambda \) is its fixed point; the two conventions are exchanged by passing to the conjugate-transposed tensor.
For a tensor \(A\) and a half-chain length \(L\), the half-chain family is the matrix \(\Psi \) with rows indexed by configurations \(\sigma \in \{ 0,\ldots ,d{-}1\} ^L\) and columns indexed by boundary pairs \((\alpha ,\beta )\), whose column at \((\alpha ,\beta )\) is the half-chain vector
The complementary family \(\Phi \) is the matrix with rows indexed by boundary pairs and columns by configurations, whose row at \((\alpha ,\beta )\) is the vector \(|\Psi _{\beta ,\alpha }\rangle \) with the two boundary indices swapped.
The coefficient kernel of the ring of \(2L\) sites is the matrix over the two half-chain configuration spaces whose entry at \((\sigma ,\tau )\) is the matrix product vector component \(V^{(2L)}(A)_{\sigma \tau }=\operatorname{tr}(A^{\sigma }A^{\tau })\) of the joined configuration.
The coefficient kernel factors as the product \(\Psi \Phi \) of the half-chain family with its complementary family. Equivalently,
Insert the resolution of the identity \(\sum _{\beta }|\beta \rangle \langle \beta |\) at the two cut bonds of the ring trace: \(\operatorname{tr}(A^{\sigma }A^{\tau }) =\sum _{\alpha ,\beta }\langle \alpha |A^{\sigma }|\beta \rangle \langle \beta |A^{\tau }|\alpha \rangle \).
The unnormalized half-chain reduced density matrix of the ring state on \(2L\) sites is \(\rho _L=KK^{\dagger }\) with \(K\) the coefficient kernel. Entrywise this is the partial trace of the unnormalized ring state over the complementary half,
and \(\rho _L\) is positive semidefinite.
The half-chain Gram matrix is \(G=\Psi ^{\dagger }\Psi \), the matrix of overlaps \(G_{(\alpha ,\beta ),(\alpha ',\beta ')} =\langle \Psi _{\alpha ,\beta } | \Psi _{\alpha ',\beta '} \rangle \) of the half-chain vectors.
The half-chain overlaps are matrix elements of the \(L\)-fold transfer map on matrix units:
Expand the overlap as a sum over configurations of products \(\langle \alpha '|A^{\sigma }|\beta '\rangle \overline{\langle \alpha |A^{\sigma }|\beta \rangle }\) and recognize the entry of \(\sum _{\sigma }A^{\sigma }|\beta '\rangle \langle \beta |(A^{\sigma })^{\dagger }\), which is the \(L\)-fold transfer map applied to the matrix unit.
Let \(S\) denote the swap of the two boundary indices. The unnormalized half-chain reduced density matrix factors exactly as
where \(SG^{\mathsf T}S\) is the Gram matrix of the complementary family.
By the splitting \(K=\Psi \Phi \) of Lemma 11.12.3, \(\rho _L=\Psi (\Phi \Phi ^{\dagger })\Psi ^{\dagger }\), and the Gram matrix \(\Phi \Phi ^{\dagger }\) of the complementary family is the transpose of the boundary-swapped half-chain Gram matrix.
The characteristic polynomials of the half-chain reduced density matrix and of the boundary comparison matrix \((SG^{\mathsf T}S)G\) of size \(D^2\) agree after matching the trivial kernel factors:
In particular the nonzero spectrum of \(\rho _L\), with multiplicities, is that of the boundary comparison matrix.
Apply the rectangular cyclicity of the characteristic polynomial to the factorization (247), regrouping the product as \(\Psi \) times \((SG^{\mathsf T}S)\Psi ^{\dagger }\); the reversed product is \((SG^{\mathsf T}S)\Psi ^{\dagger }\Psi =(SG^{\mathsf T}S)G\).
Let \(A\) have trace-preserving transfer map \(\mathcal{E}_A\) with fixed point \(\Lambda \), \(\operatorname{tr}(\Lambda )\neq 0\), and suppose
where \(P_{\Lambda }\) is the fixed-point projection. Then there are constants \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that for every \(L\geq 1\) and all boundary pairs,
The hypothesis (249) is the spectral gap condition of [ 228 , Theorem 6 ] in the trace-preserving gauge: apart from the simple eigenvalue one, the spectrum of \(\mathcal{E}_A\) lies strictly inside the unit disc. The rate \(r\) is chosen strictly between the spectral radius of \(\mathcal{E}_A-P_{\Lambda }\) and one.
Write \(\mathcal{E}_A^{L}=P_{\Lambda }+(\mathcal{E}_A-P_{\Lambda })^{L}\) for \(L\geq 1\) by the power decomposition. By (246) the deviation of a Gram entry from the fixed-point projection entry is an entry of \((\mathcal{E}_A-P_{\Lambda })^{L}\) applied to a matrix unit, bounded by the operator norm of the \(L\)-th power, which decays geometrically below unit spectral radius.
For a diagonal fixed point \(\Lambda =\operatorname {diag}(\lambda _1,\ldots ,\lambda _D)\) with \(\operatorname{tr}(\Lambda )\neq 0\), the limiting overlaps form a diagonal matrix in the boundary pairs:
For \(\operatorname{tr}(\Lambda )=1\) this is the display \(\lambda _\alpha \delta _{\alpha ,\alpha '}\delta _{\beta ,\beta '}\) of [ 228 ] .
Evaluate the fixed-point projection on the matrix unit: its trace is \(\delta _{\beta ,\beta '}\), and the diagonal entry of \(\Lambda \) at \((\alpha ',\alpha )\) is \(\lambda _\alpha \delta _{\alpha ,\alpha '}\).
Let \(A\) have trace-preserving transfer map with diagonal fixed point \(\Lambda =\operatorname {diag}(\lambda _1,\ldots ,\lambda _D)\), \(\operatorname{tr}(\Lambda )\neq 0\), satisfying the spectral condition (249). Then there are constants \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that for every \(L\geq 1\) the boundary comparison matrix satisfies, entrywise,
The limiting matrix is the diagonal matrix \(\Lambda \otimes \Lambda \) up to the trace normalization. Together with Theorem 11.12.8, the nonzero spectrum of the half-chain reduced density matrix on the ring of \(2L\) sites is that of a \(D^2\times D^2\) matrix converging entrywise, geometrically in \(L\), to \(\Lambda \otimes \Lambda /\operatorname{tr}(\Lambda )^2\). Theorem 11.12.1.7 proves convergence of the normalized physical eigenvalues.
Both factors of the comparison matrix are entrywise within \(Cr^{L}\) of their diagonal limits by Theorem 11.12.9 and Lemma 11.12.10, the swapped-transposed factor because swapping and transposing permute entries and exchange the two diagonal weights. The limit entries are bounded, so the product deviates from the product of the limits by at most a constant multiple of \(r^{L}\), with constant \(D^2\bigl(C(B+C)+BC\bigr)+1\) for an entry bound \(B\) on the limits.
11.12.1 Ordered eigenvalues and finite-size normalization
For a Hermitian matrix \(M\) of size \(n\), let \(s_k(M)\) be its \(k\)th largest eigenvalue, counted with multiplicity, for \(0\leq k{\lt}n\), and set \(s_k(M)=0\) for \(k\geq n\). If \(M\geq 0\), all these values are nonnegative.
An increasing real function acts termwise on the ordered eigenvalues of a Hermitian matrix. In particular, adding \(tI\) adds \(t\) to each eigenvalue, and multiplication by \(t\geq 0\) multiplies each eigenvalue by \(t\). If \(M\) and \(N\) are Hermitian matrices of the same size, then \(|s_k(M)-s_k(N)|\leq \lVert M-N\rVert _{\mathrm{op}}\) for every \(k\). Consequently, convergence of Hermitian matrices of fixed size implies convergence of every zero-extended ordered eigenvalue.
Apply the scalar function to the spectral decomposition and sort the characteristic roots in decreasing order. For the perturbation bound, use \(-\lVert M-N\rVert I\leq M-N\leq \lVert M-N\rVert I\), Weyl monotonicity, and the scalar-translation identity. The resulting bound implies convergence.
Let \(M\geq 0\) have size \(n\) and \(N\geq 0\) have size \(m\). If \(X^m\chi _M(X)=X^n\chi _N(X)\), then \(s_k(M)=s_k(N)\) for every \(k\geq 0\).
Sort the real characteristic roots. Positivity places the additional \(m\) and \(n\) zeros at the ends of the respective ordered lists. Extending either list by further zeros therefore gives the same sequence.
Write \(W_L=\Phi _L\Phi _L^\dagger =SG_L^{\mathsf T}S\) and define
Here \(\rho _L\) is the actual unnormalized physical reduced matrix. Both normalized matrices are defined as zero when \(Z_L=0\). The physical entanglement spectrum is \(s_k(\widehat\rho _L)\).
For every finite half-chain length \(L\), \(H_L\geq 0\) and
The same characteristic-polynomial identity holds before normalization. Both normalized matrices are positive semidefinite, and \(s_k(\widehat\rho _L)=s_k(\widehat H_L)\) for every \(k\). In particular, the physical spectrum vanishes for \(k\geq D^2\). These statements include \(d^L{\lt}D^2\), singular Gram matrices, and zero-norm lengths. If \(Z_L{\gt}0\), then \(\operatorname {tr}\widehat\rho _L=1\).
The matrix \(H_L\) is a positive congruence of \(W_L\). Cyclicity of trace and \(\sqrt{G_L}\sqrt{G_L}=G_L\) give (255). Apply the rectangular-product characteristic-polynomial identity to \(\rho _L=\Psi _L W_L\Psi _L^\dagger \) and then the square-product identity to \(H_L\). Insert the same scalar \(Z_L^{-1}\) into both products to obtain (256), and apply Theorem 11.12.1.3.
For \(\lambda _\alpha \geq 0\) with \(\sum _\alpha \lambda _\alpha =1\), let \(\Lambda =\operatorname {diag}(\lambda _\alpha )\), \(G_\infty =\operatorname {diag}_{\alpha \beta }(\lambda _\alpha )\), and \(H_\infty =\Lambda \otimes \Lambda =\operatorname {diag}_{\alpha \beta }(\lambda _\alpha \lambda _\beta )\). The matrix \(H_\infty \) is positive semidefinite and has trace one.
Suppose \(\mathcal{E}_A\) is trace preserving, fixes \(\Lambda \), and satisfies (249). Then \(Z_L\to 1\), so \(Z_L{\gt}0\) for all sufficiently large \(L\). The actual normalized physical eigenvalues obey
Repeated eigenvalues are counted with multiplicity, and both lists use the zero-extension convention of Definition 11.12.1.1. No invertibility of \(\Lambda \) or finite-length normalization is assumed.
The geometric transfer estimate \(Cr^L\) gives \(G_L\to G_\infty \). More generally, if \(G_L\to \operatorname {diag}(g_p)\) with \(g_p\geq 0\), continuity of the positive square root gives \(H_L\to \operatorname {diag}(g_p g_{p^{\mathrm{swap}}})\). Apply this with \(g_{\alpha \beta }=\lambda _\alpha \) to obtain \(H_L\to \Lambda \otimes \Lambda \). Taking traces gives \(Z_L\to 1\), hence \(\widehat H_L\to \Lambda \otimes \Lambda \). Weyl’s bound and the exact physical comparison yield (257). This use of square-root continuity asserts convergence, not a sharper quantitative spectral rate.
11.12.2 The unital canonical convention
For \(B_i=A_i^\dagger \), the half-chain Gram matrices satisfy \(G_L(B)_{(\beta ',\alpha '),(\beta ,\alpha )} =G_L(A)_{(\alpha ,\beta ),(\alpha ',\beta ')}\) at every length.
The transfer map of \(B\) is the trace-pairing adjoint of that of \(A\). The adjoint of a power is the corresponding power of the adjoint. Apply this identity to the matrix units in the overlap formula.
Let \(A\) be a normal tensor in the unital canonical convention, \(\sum _i A_iA_i^\dagger =I\), and let \(\Lambda =\operatorname {diag}(\lambda _\alpha )\geq 0\) have trace one and satisfy \(\sum _i A_i^\dagger \Lambda A_i=\Lambda \). Then \(Z_L\to 1\) and \(Z_L{\gt}0\) for all sufficiently large \(L\). For every \(k\in \mathbb N\), the actual physical reduced density obeys \(s_k(\widehat\rho _L)\to s_k(\Lambda \otimes \Lambda )\). This statement assumes normality explicitly. The source canonical conditions and Condition C2 are treated in Theorem 11.12.2.3. The index and decay conventions are detailed in [ 63 ] . The intermediate Gram limit is \(\operatorname {diag}_{\alpha \beta }(\lambda _\beta )\).
Normality gives irreducibility and a unique peripheral eigenvalue. The dual channel has the same eigenvalues under the trace pairing; irreducibility also passes to the adjoint family. Its complementary fixed-point map therefore has spectral radius strictly below one. Apply the geometric Gram estimate to \(B_i=A_i^\dagger \) and use Theorem 11.12.2.1 to return to the original tensor \(A\). With \(g_{\alpha \beta }=\lambda _\beta \), the positive boundary limit has entries \(g_{\alpha \beta }g_{\beta \alpha } =\lambda _\alpha \lambda _\beta \). The norm and ordered-eigenvalue arguments of Theorem 11.12.1.7 now apply to the original physical reduced matrices.
Let \(A\) satisfy the three one-block canonical conditions of [ 228 , Theorem 4 ] : \(\sum _i A_iA_i^\dagger =I\), a faithful diagonal dual fixed point \(\Lambda =\operatorname {diag} (\lambda _\alpha ){\gt}0\), and \(\{ X:\mathcal{E}_A(X)=X\} =\mathbb C I\). Normalize \(\operatorname {tr}\Lambda =1\) and the transfer spectral radius to one. Assume Condition C2: every unit-modulus transfer eigenvalue is one. Then \(A\) is normal. For every \(k\in \mathbb N\), the ordered physical half-chain eigenvalues, with multiplicities and zero extension, satisfy \(s_k(\widehat\rho _L)\to s_k(\Lambda \otimes \Lambda )\). This gives the one-block canonical half-chain conclusion of [ 228 , Theorem 6 ] without an additional normality or complementary-gap assumption. The hypothesis that this is a single canonical block is explicit; reduction of a general weighted canonical representation is not asserted here.
The faithful dual fixed point and scalar fixed-point condition imply irreducibility: an invariant projection has a positive defect with zero pairing against \(\Lambda \), hence is fixed, scalar, and trivial. Unitality supplies the identity eigenvector, while Condition C2 excludes other peripheral eigenvalues. These statements and the spectral-radius normalization give normality. Apply Theorem 11.12.2.2 to the original tensor.
For a TI MPS satisfying Condition C2, the source asserts convergence of the half-chain reduced spectrum to the doubled canonical fixed-point spectrum [ 228 , Theorem 6 ] . For a general weighted canonical representation, passing to its dominant block and controlling the subleading physical contribution remains separate from Theorem 11.12.2.3; see [ 63 ] .
11.13 Two-block uncle Hamiltonians as actual parent limits
The uncle construction takes a punctured perturbation limit of the canonical parent Hamiltonian. The binding source is [ 196 , continuity-of-projector lemma, span-invariance lemma, and form theorem ] ; these are in Papers/1210.6613/UncleHMPS.tex, lines 314–340, 890–902, and 912–953. The word “gauge” in the span lemma allows an invertible linear map on virtual matrices. In particular, multiplying both off-diagonal blocks by one scalar need not be a similarity transformation.
Let \(T_x\) be a family of finite-dimensional boundary maps continuous at \(x_0\). If \(T_{x_0}\) is injective, its range projector is continuous at \(x_0\). Consequently, fix a window length \(L\) and a tensor family continuous at \(x_0\). If the products of exactly \(L\) tensor matrices at \(x_0\) span the full virtual matrix algebra, the canonical parent interaction on \(L\) sites is continuous at that point.
Injectivity is open. In a neighborhood of \(x_0\), write the projector as \(T_x(T_x^\dagger T_x)^{-1}T_x^\dagger \) using the existing inverse-Gram formula. Inversion is continuous at the invertible base-point Gram operator. This is the continuous-basis argument of the source, with inverse Gram matrices in place of explicit Gram–Schmidt coordinates.
For \(z\ne 0\), the linear map \(\mathcal L_z\bigl(\begin{smallmatrix} X & Y \\ Z & W \end{smallmatrix}\bigr) =\bigl(\begin{smallmatrix} X & zY \\ zZ & W \end{smallmatrix}\bigr)\) is invertible and preserves the one-site physical span of a tensor.
Its inverse is \(\mathcal L_{z^{-1}}\). The bilinear trace identity \(\operatorname{tr}(\mathcal L_z(M)X)=\operatorname{tr}(M\mathcal L_z(X))\) transfers the rescaling to the arbitrary boundary matrix. Surjectivity of this boundary change gives equality of the physical ranges. No conjugation or similarity assumption is used.
Let \(A_i,B_i\) be square virtual blocks, and let \(P_i,Q_i,R_i,L_i\) be arbitrary perturbation blocks of the matching sizes. Define
The regular interpolant \(U(\varepsilon )\) has diagonal blocks \((A_i+\varepsilon P_i)(A_j+\varepsilon P_j)+\varepsilon ^2R_iL_j\) and \((B_i+\varepsilon Q_i)(B_j+\varepsilon Q_j)+\varepsilon ^2L_iR_j\); its off-diagonal blocks are those of \(U\) plus respectively \(\varepsilon (P_iR_j+R_iQ_j)\) and \(\varepsilon (L_iP_j+Q_iL_j)\). In particular \(U(0)=U\).
The interpolant is continuous, and for every \(\varepsilon \ne 0\), \(U(\varepsilon )=\mathcal L_{\varepsilon ^{-1}} (C(\varepsilon )\mathbin {\circ }C(\varepsilon ))\).
Multiply the four matrix blocks and cancel the nonzero factor \(\varepsilon \) in each off-diagonal block. All remaining entries are polynomials in \(\varepsilon \).
Suppose the explicit uncle tensor \(U\) is injective. The actual canonical two-site parent interactions of \(C(\varepsilon )\) converge, in operator norm as \(\varepsilon \to 0\) through nonzero complex parameters, to \(h'_P=\Pi [U]\) on the original two physical sites. At every fixed periodic length \(N\ge 2\), the actual parent Hamiltonians converge in operator norm to \(H'_P=\sum _{i=1}^N(h'_P)_{i,i+1}\). The Hamiltonian limit also holds along real perturbation parameters.
For nonzero parameters, virtual rescaling preserves the physical range and hence the canonical projector. The regular interpolant has injective value \(U\) at zero, so its projector converges to \(\Pi [U]\). The existing physical-blocking isometry returns this projector to the original two-site Hilbert space. Each translated interaction is expressed through the existing cyclic restriction and its adjoint. Finite sums and these fixed operator compositions preserve operator-norm convergence.
The source assumes \(C=A\oplus B\) is block-injective. The local limit proof needs only the stated injectivity of \(U\), so the formal result also covers other block data satisfying that condition as an algebraic operator-limit statement, without additional claims about their standard-form physical interpretation. Arbitrary diagonal perturbations \(P,Q\) have the same limit for fixed \(A,B,R,L\). No assertion about generic perturbations, an algebraic exceptional set, gaplessness, or an infinite-volume Hamiltonian is included here.