Dependencies

If the parent-ground-space dimension is eventually constant, then for every \(N{\gt}2\) it equals the number of positive loops in the sector graph. In particular, this conclusion holds under the BNT and all-chain nearest-neighbor parent-ground-space hypotheses of Theorem 13.1.71.

For an Appendix B structural datum and a periodic chain of length \(N{\gt}2\), set \(p_i=h_i^{(N)}\) for \(i\in \mathbb Z/N\mathbb Z\), where \(h_i^{(N)}\) is the translated length-two parent interaction. These maps are idempotent and commute pairwise: adjacent commutation follows from Theorem 13.1.61, while disjoint length-two interactions commute by locality.

Suppose that \(A\) is equipped with Appendix B structural data \((X,\Lambda ,U)\): the matrix \(X\) is invertible, the diagonal entries of \(\Lambda \) are strictly positive, \(A^i=X\Lambda U^iX^{-1}\), and \(U\) satisfies the pair-index isometry equation below. Let \(\mathcal H_a\) and \(\mathcal H_b\) have bases indexed by \(\alpha ,\beta \in \{ 0,\ldots ,D-1\} \). The coefficients \(U^i_{\alpha ,\beta }\) define maps

and

For every \(N\geq 0\), the coefficients of \(U\) define

with

The conjugate coefficient map is

This is the tensor power occurring in the basic-vector expression [ CPGSV16 , Equation (3.17) ] .

The \(AX\) coefficient-space representative is, before the \(AX\)-lift, the canonical two-site parent interaction of the core tensor, \(\widehat Q_{AX}=q_2(A_{\mathrm{core}})\). The source projector \(Q_{AX}\) of Definition D.2 acts on \(\mathcal H_A\otimes \mathcal H_X\). The displayed operator is the two-site coefficient representative that will be placed on the \(AX\) face; it is not by itself the construction of that source projector.

The \(XB\) coefficient-space representative is, before the \(XB\)-lift, the same coefficient-space representative, \(\widehat Q_{XB}=\widehat Q_{AX}\). The source projector \(Q_{XB}\) of Definition D.2 acts on \(\mathcal H_X\otimes \mathcal H_B\). The displayed operator is the coefficient representative later placed on the \(XB\) face.

For Appendix B structural data, the two-site basic support is \(G_2(A_{\mathrm{core}})=G_2(\Lambda U)\).

Let \(P_2\) be the orthogonal projector onto \(G_2(\Lambda U)\), as in the local parent-space construction of [ CPGSV16 , Definition 3.9 ] .

Put \(|\varphi \rangle =\sum _b\lambda _b|b,b\rangle \). For outer-boundary coefficients \(v_{a,c}\), insertion of this vector on the middle virtual bond is the map \(I_\varphi \) defined by

The two-site physical map is \(U^{\otimes 2}I_\varphi \). A boundary matrix \(Y\) determines outer-boundary coefficients by \((v_Y)_{a,c}=\lambda _aY_{c,a}\). Opening the outer virtual bond in the two-site contraction of [ CPGSV16 , Equations (3.17)–(3.18) ] gives these coefficients.

Write \(s=\langle \varphi ,\varphi \rangle =\sum _b\lambda _b^2\). The orthogonal projector onto \(\mathbb {C}\varphi \) has pair-index coefficients

On three virtual site pairs, let \(P_{01}\) place this projector on the bond joining the right index of site zero to the left index of site one, and let \(P_{12}\) place it on the next bond.

Let \(a\to a\) be a positive loop and choose a nonzero vector \(\varphi _a\in \operatorname{ran}P_{a,a}\). In sector coordinates define the matrix product tensor \((C_a^{(r,l)})_{x,y}=\delta _{x,l}\varphi _a(r,y)\) and set it to zero outside sector \(a\). The physical loop tensor is the image of \(C_a\) under the one-site unitary of the spatial decomposition. Its associated cyclic product pairs the right factor at site \(n\) with the left factor at site \(n+1\). The corresponding \(N\)-site product state is denoted by \(\Psi _a^{(N)}\in (\mathbb {C}^d)^{\otimes N}\).

Let \(\Phi _a=(\varphi _a(r,l))_{r,l}\) and write \(r_a=\operatorname{rank}\Phi _a\). Choose a rank factorization

The tensor on the minimal Schmidt support is

again extended by zero outside sector \(a\). Its physical tensor is its image under the same one-site unitary.

Let \(\nu _a=\lVert \varphi _a\rVert _2{\gt}0\), where the norm is the Hilbert-space norm of the bond vector, or equivalently the Frobenius norm of \(\Phi _a\). In either coordinate system, the normalized representative is \(\widehat C_a=\nu _a^{-1}\widetilde C_a\). The raw coordinate and physical tensors are retained separately.

Suppose that a one-site unitary identifies the physical Hilbert space with a finite direct sum

and that in these coordinates the complementary two-site interaction is \(\bigoplus _{a,b}\mathbb {1}_{a,l}\otimes P_{a,b}\otimes \mathbb {1}_{b,r}\), where each \(P_{a,b}\) is an orthogonal projector. The directed edge \(a\to b\) is present precisely when \(\operatorname{ran}P_{a,b}\) is nonzero. An ordered cycle of length \(N\) is a sequence \((a_0,\ldots ,a_{N-1})\) for which every cyclically adjacent edge \(a_n\to a_{n+1}\) is present. Cyclic rotations are not identified. This is the graph following equation (2) of [ Bei12 , Section III ] .

Given a tensor \(A\) and a blocking length \(L\), the \(L\)-blocked tensor is the tensor with physical index set \(\{ 0,\ldots ,d{-}1\} ^L\) (hence physical dimension \(d^L\)) and the same bond dimension \(D\), whose matrices are indexed by words \((i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\):

Blocking coarse-grains \(L\) neighbouring sites into one tensor: the inner virtual bonds are contracted, the \(L\) physical legs merge into a single composite index, and the two outer bonds remain as the bond of \(A^{[L]}\).

A basis of normal tensors (BNT) for a total tensor \(A_{\mathrm{tot}}\) consists of \(g\) block tensors \((A_1, \ldots , A_g)\) with bond dimensions \((D_1, \ldots , D_g)\) such that each \(A_j\) is normal (Definition 2.4.4); at each system size \(N\geq 1\) the MPV of \(A_{\mathrm{tot}}\) lies in the span of the block MPVs \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\); and for sufficiently large \(N\) those block MPVs are linearly independent. Here normality means eventual block injectivity, as in Definition 2.4.4. Thus this is the algebraic form of [ CPGSV21 , Definition IV.2 ] , rather than its primitive-channel formulation. In the canonical-form characterization of [ CPGSV16 , Section 2.3 ] , a BNT is a minimal family of representatives for \(\widetilde A_k^i=e^{i\phi _k}X_kA_j^iX_k^{-1}\) among the normal blocks.

For a basis of normal tensors \(A_1,\ldots ,A_g\), the vector space generated by their length-\(N\) matrix product vectors is \(\operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \).

Let \(A\) be an MPS tensor, let \(\Omega _K\) be the set of length-\(K\) words in the physical alphabet, and let \(\mathcal B_K=\bigoplus _{u\in \Omega _K}M_{D}(\mathbb {C})\) with its Hilbert–Schmidt direct-sum norm. Define

by applying the one-site left virtual map independently to each prefix fiber. Define

by applying the length-\(K\) tail virtual map independently to each final-site fiber. These definitions make sense without any nonemptiness assumption on the virtual dimension or on either configuration set.

When \(D\geq 1\) and \(\rho \in M_{D}(\mathbb {C})\) is positive definite, set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and define the centered virtual residual by

Let \(R_{K,l}\) be the centered virtual residual and let \(S_n\) be the inverse Gram at length \(n\). The centered physical sandwich is

Assume \(D\geq 1\) and let \(\rho \in M_{D}(\mathbb {C})\) be positive definite. Set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). Suppose that \(\Gamma _{l+1}^{\mathrm{ES}}\), \(\Gamma _{K+l}^{\mathrm{ES}}\), and \(\Gamma _{K+l+1}^{\mathrm{ES}}\) are injective, and write

The three correction sandwiches in Theorem 13.3.3.25 are

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 [ CPGSV21 , 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.

The Hilbert-space cyclic constraint space is the transport of \(G_L^{(N)}(A)\) by the canonical \(\ell ^2\) identification: \(G_{L,\mathrm{ES}}^{(N)}(A)=e_N^{-1}G_L^{(N)}(A)\).

If \(L\leq N\) and \(i\in \{ 0,\ldots ,N{-}1\} \), identify the sites of the periodic chain with the disjoint union of the cyclic window \(i,i+1,\ldots ,i+L-1\) and its ordered complement \(i+L,i+L+1,\ldots ,i+N-1\), with all indices taken modulo \(N\).

For a length-\(L\) cyclic window on \(\{ 0,\ldots ,N{-}1\} \) starting at \(i\), its support is \(S_i=\{ i,i+1,\ldots ,i+L-1\} \pmod N\), with repeated visits counted once.

Two length-\(L\) cyclic windows overlap, written \(i\sim _L j\), when their supports meet: \(S_i\cap S_j\ne \varnothing \).

Given a word \(w = (i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\), the word evaluation is the matrix product

The empty word evaluates to the identity, \(A^\varnothing = \mathbb {1}_D\). In tensor-network notation,

in which each black node denotes the same local tensor \(A\), the virtual legs remain open, and the physical legs are labelled by the word \((i_1,\ldots ,i_L)\).

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 \(0\le r{\lt}L\).

Let \(U\) and \(V\) be subspaces of a finite-dimensional complex Hilbert space. Their Friedrichs overlap is at most \(\eta \) when

for every \(x\in U\cap (U\cap V)^\perp \) and \(y\in V\cap (U\cap V)^\perp \).

Two tensors \(A\) and \(B\) of the same bond dimension \(D\) are gauge equivalent if there exists an invertible matrix \(X \in \mathrm{GL}_D(\mathbb {C})\) such that, for every \(i \in \{ 0,\ldots ,d{-}1\} \),

In tensor-network notation, (5) is represented by

where the black node denotes the tensor \(A\), the upper leg is the physical index \(i\), and the two red side nodes denote the gauge matrices acting on the virtual legs.

Two tensors \(A\) and \(B\) are gauge-phase equivalent if there exist \(X \in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\setminus \{ 0\} \) such that, for every physical index \(i\),

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 \(((c,e),(a,b))\).

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 \(G_L^{\mathrm{ES}}(A):=\iota _L(G_L(A)) \subseteq \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\).

The boundary Gram operator is

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,

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 \(L\).

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 \(M_{D}(\mathbb {C})\).

The local ground space is the image \(G_L(A):=\operatorname{range}(\Gamma _L)\subseteq (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\).

The structure used below records only the idempotent-product consequence of the parent commuting Hamiltonian condition of [ CPGSV16 , Appendix D.2 ] . It consists of commuting idempotent endomorphisms \(P_{AX}\) and \(P_{XB}\) such that, for the idempotent \(P_K\) associated to \(K\), \(P_{AX}\circ P_{XB}=P_K\). In the source, this is the equation obtained from local projectors \(Q_{AX}\) and \(Q_{XB}\) satisfying \([Q_{AX},Q_{XB}]=0\) and \(K_{AXB}=(K_{AX}\otimes H_B)\cap (H_A\otimes K_{XB})\).

For a fixed chain length \(N\), this condition is the conjunction of the nearest-neighbor commutation equation, the zero-energy equation for the MPS vector, and the ground-space spanning equation:

The source statement in [ CPGSV16 , Theorem 3.10(iii) ] quantifies this condition over all \(N{\gt}2\).

This is the source quantification in [ CPGSV16 , Theorem 3.10(iii) ] . For every \(N{\gt}2\), the fixed-chain condition of Definition 13.1.17 holds:

This condition records the ground-space spanning equation, but does not prove it for a concrete canonical-form tensor.

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

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 \(T^\dagger T\).

The parent Hamiltonian with block length \(L\) on \(N\) sites has commuting local terms if \(h_i h_j=h_j h_i\) for all \(i,j\in \{ 0,\ldots ,N{-}1\} \). This records the translated commutation equation \([\tau _j(P_L),P_L]=0\) from [ CPGSV16 , Definition 3.9 ] ; the same source definition also requires the ground space to be spanned, for \(N{\gt}L\), by the periodic MPS vectors of the BNT components.

Given an endomorphism \(P_K\) and sets of observables \(\mathcal{O}_A\), \(\mathcal{O}_B\), regions \(A\) and \(B\) are decorrelated w.r.t. \(K\) when \(P_K\circ O_A\circ (\mathbb {1}-P_K)\circ O_B\circ P_K=0\) for all \(O_A\in \mathcal{O}_A\) and \(O_B\in \mathcal{O}_B\).

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 \(i\in \{ 0,\ldots ,N{-}1\} \).

The nearest-neighbor commuting condition is the length-\(2\) specialization of the preceding commutation equation. For every \(i,j\in \{ 0,\ldots ,N{-}1\} \), one has \(h_i h_j=h_j h_i\). The source definition [ CPGSV16 , Definition 3.9 ] also requires the corresponding parent Hamiltonian to have \(L=2\) and to have the prescribed ground-space spanning property for \(N{\gt}2\).

This condition records the commutation equation and the zero-energy equation for the nearest-neighbor local terms:

This is the zero-energy part of the condition appearing in [ CPGSV16 , Theorem 3.10(iii) ] : the MPS vector has zero energy for the translated two-site local terms, and those terms commute. The missing source assertion is that the whole parent-Hamiltonian ground space is spanned by the corresponding periodic MPS vectors from the BNT components, as recorded in Definition 13.1.16.

A family \((U_j)_j\) satisfies the joint isometry condition when

Each block satisfies the within-block pair-index orthonormality, and the cross-block sums between distinct blocks vanish. This is the isometry condition of [ CPGSV16 , Theorem 3.11 ] , split into its \(j=j'\) and \(j\ne j'\) cases.

A tensor \(A\) is \(L\)-block injective if

An injective tensor is \(1\)-block injective.

For \(K,L\in \mathbb {N}\), the left boundary map is the \(\mathbb {C}\)-linear map

defined for every family \(Z\colon (\{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} )\to M_{D}(\mathbb {C})\) by

where \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the prefix and \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the suffix. When \(Z_\tau =A^\tau X\) the output equals the standard \(\Gamma _{K+L}(X)\).

For each site index \(i\in \{ 0,\ldots ,N{-}1\} \), the translated local term is determined by

The transfer map of \(A\) satisfies the idempotence criterion when \(\mathcal{E}_A\circ \mathcal{E}_A=\mathcal{E}_A\).

The subspace spanned by the MPS vector \(V^{(N)}(A)(\sigma )=\operatorname{tr}(A^{\sigma _0}\cdots A^{\sigma _{N-1}})\).

A tensor \(A\) is normal [ CPGSV21 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

For local dimension \(d\) and chain length \(N\), the \(N\)-site Hilbert space is \(\mathcal H_{N}^{(d)}:=\{ 0,\ldots ,d{-}1\} ^{N}\to \mathbb {C}\). This is the computational-basis identification used throughout the chapter for periodic-chain states.

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of  [ CPGSV16 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

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.

For \(L,N\in \mathbb {N}\), let

The compatible length-\(N\) open-chain parent Hamiltonian of interaction length \(L\) is

Thus no summand crosses the two endpoints of the interval. For \(L=l+1\), the starting sites are \(0,\ldots ,N-l-1\), in agreement with [ Nac96 , Equation (3.12) ] .

On a three-site space \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\), let \(Q_{AX}\) and \(Q_{XB}\) be two-site projectors. Their local actions on the three-site space are the operators

The algebraic part of the overlapping nearest-neighbor commutation condition is

after these two local actions are viewed on the common three-site space. This is the local-support part of the parent commuting Hamiltonian condition in [ CPGSV16 , Definition D.2 ] . The cyclic windows beginning at \(i\) and \(i+1\) realize the two supports \(AX\) and \(XB\), and hence overlap at the middle site.

The source local condition also fixes a three-site subspace \(K_{AXB}\) and imposes the kernel-intersection equation \(K_{AXB}=\ker ((Q_{AX})_{AX})\cap \ker ((Q_{XB})_{XB})\). Forgetting this intersection equation gives the algebraic overlapping two-site commutation condition displayed above.

The two lift maps also distribute over complementary projectors:

Consequently, if the projectors \(Q_{AX}\) and \(Q_{XB}\) commute on the three-site space, then their complementary projectors \(P_{AX}=1-Q_{AX}\) and \(P_{XB}=1-Q_{XB}\) commute after the same lifts. This is the elementary algebraic passage used when one moves between the \(Q\)-projectors of [ CPGSV16 , Appendix D.2 ] and the complementary ground-space projectors \(P\). Since the original two projectors are idempotent, the complementary projectors are idempotent as well; hence the pair \(P_{AX},P_{XB}\) satisfies the same idempotent-and-commutation condition on the three-site space.

Denote by \(\mathcal G_{N,L}^{\mathrm{ES}}(A)\) the parent ground space after identifying the coefficient space with the Hilbert space \(\ell ^2(\{ 0,\ldots ,d-1\} ^N)\).

The chain Hamiltonian is

Denote by \(H_{N,L}^{\mathrm{ES}}(A)\) the conjugate of \(H_N(A,L)\) by the canonical identification between the coefficient space and \(\ell ^2(\{ 0,\ldots ,d-1\} ^N)\).

Let \(B\) be a canonical-form tensor with BNT components \(A_1,\ldots ,A_g\). The source parent-Hamiltonian spanning condition is that, for every \(N{\gt}L\), \(\ker H_L^{(N)}(B)=\operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \). This is the ground-space clause in [ CPGSV16 , Definition 3.9 ] . It is separate from the commutation equation \([\tau _j(P_L),P_L]=0\) and from the zero-energy equation for a single periodic MPS vector.

The parent interaction at range \(L\) is the canonical orthogonal projector specified in (2).

For \(K,L\in \mathbb {N}\) and a fixed suffix configuration \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \), the prefix restriction is the \(\mathbb {C}\)-linear map \(P_\tau \colon \mathbb {C}^{\{ 0,\ldots ,d{-}1\} ^{K+L}}\to \mathbb {C}^{\{ 0,\ldots ,d{-}1\} ^K}\) defined by \((P_\tau \psi )(\sigma )=\psi (\sigma ,\tau )\) for \(\sigma \colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \). This is the left-sided analogue of the suffix restriction of Definition 12.4.5.5.

Assume \(D{\gt}0\). A tensor \(A\) together with a matrix \(\rho \in M_{D}(\mathbb {C})\) is primitive if

where \(P\) is the fixed-point projection associated to \(\rho \). When the choice of \(\rho \) is irrelevant, we simply say that \(A\) is a primitive MPS tensor. This condition combines a complementary spectral gap with a nonzero positive semidefinite fixed point. It is not the paper definition in Definition 7.1.1.3, which is the uniform spreading condition (1). With the additional hypothesis \(\rho {\gt}0\), the complementary-gap condition implies strong irreducibility and hence paper primitivity by Theorem 7.1.1.13.

Define the tail boundary map in the common \((K+L+Q)\)-site coordinates by

Let \(\mathcal V=\bigsqcup _j\{ j\} \times [D_j]\times [D_j]\) be the disjoint union of the within-sector virtual pairs, and let \(V\colon \mathcal V\to \mathbb {C}^d\) be the common physical map whose matrix elements are \(V_{i,(j,\alpha ,\beta )}=(U_j^i)_{\alpha ,\beta }\). For each sector insert the vector \(|\varphi _j\rangle =\sum _b\lambda _{j,b}|b,b\rangle \) on the middle virtual bond, apply \(V\) on both sites, and sum the resulting physical vectors. This defines the equal-sector two-site embedding

Unequal-sector virtual bonds do not occur. The corresponding physical support operator \(P_{2,\mathrm{res}}\) is the orthogonal projector onto \(G_2(\bigoplus _j B_j)\).

Fixing the first physical index \(i\) gives the state \(\sigma \mapsto \psi (i,\sigma _1,\ldots ,\sigma _L)\).

Fixing the last physical index \(j\) gives the state \(\sigma \mapsto \psi (\sigma _1,\ldots ,\sigma _L,j)\).

Given a sector decomposition \(P\), these conditions define the normalized, left-canonical, aperiodic BNT surface used below. This is stronger than the abstract BNT of Definition 9.1.1 and the literal coefficient-form BNT of Definition 9.1.2: besides spanning and eventual linear independence, it fixes irreducible left-canonical representatives, their self-overlap normalization, gauge-phase separation, and the canonical-form weight normalization. It imposes no equal-modulus factorization or strict ordering on the raw weights. Explicitly, each basis block has positive bond dimension, is irreducible, and is left-canonical; the normalized self-overlap of every basis block converges to \(1\); the basis MPV family is eventually linearly independent; distinct same-dimension basis blocks are not gauge-phase equivalent after identifying equal bond dimensions; and the [ CPGSV16 , Section 2.3 ] canonical-form normalization holds,

The MPV expansion takes the raw two-layer form

matching [ CPGSV16 , Section 2.3 ] and [ CPGSV21 , Definition IV.2 ] . The canonical-form normalization admits every example of the source paper (in particular \(C \oplus D\), \(C \oplus (-C)\), \(C \oplus (1/2) C\), and \(C \oplus (-C) \oplus (1/2) C\)) and is not derivable from the per-block normality hypotheses alone.

A sector-weight tuple over \(g\) basis blocks consists of multiplicities \(r_j {\gt} 0\), nonzero weights \(\mu _{j,q}\) for \(q \in \{ 1,\ldots ,r_j\} \), and the sector coefficients

A sector decomposition consists of the basis blocks \((A_j)_{j=1}^g\) together with these multiplicities and weights. If \(A_j\) has bond dimension \(D_j\), then the flattened copy family has total bond dimension

Transport the tail and left boundary maps through the canonical Euclidean identifications on their virtual matrix families and physical coefficient spaces. Denote the resulting continuous maps by \(\Gamma ^{\mathrm{tail}}_{K,L}\) and \(\Gamma ^{\mathrm{left}}_{K,L}\).

For \(K,L\in \mathbb {N}\), the tail boundary map is the \(\mathbb {C}\)-linear map

defined for every family \(Y\colon (\{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} )\to M_{D}(\mathbb {C})\) by

where \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the prefix and \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \) is the suffix. When \(Y_u=X A^u\) the output equals the standard \(\Gamma _{K+L}(X)\).

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 \((R^{\mathrm{tail}}_u\psi )(\sigma )=\psi (u,\sigma )\).

For every type \(\alpha \), evaluation at the three coordinates gives the canonical equivalence

Its forward map is

The complex-linear extension requires \(P O_A(\mathbb {1}-P)O_B P=0\) for arbitrary complex matrices \(O_A\) and \(O_B\).

Writing \(\widehat M_R\) for the lift of an operator \(M_R\) supported on \(R\in \{ AX,XB,A,B\} \), the four lifts to \(H_A\otimes (H_X\otimes H_B)\) are

The first and fourth formulas use the canonical identification \((H_A\otimes H_X)\otimes H_B\cong H_A\otimes (H_X\otimes H_B)\).

For a Hermitian operator \(P\), let \(P_{AX}\) and \(P_{XB}\) be the support projectors of \(\operatorname{tr}_B P\) and \(\operatorname{tr}_A P\), respectively, as in [ CPGSV16 , Appendix D.2, lines 2225–2229 ] . Write \(\widehat P_{AX}\) and \(\widehat P_{XB}\) for their lifts to the tripartite space.

The two reduced operators are \(\rho _{AX}=\operatorname{tr}_B P\) and \(\rho _{XB}=\operatorname{tr}_A P\).

Regions \(A\) and \(B\) are decorrelated relative to \(P\) if, for all Hermitian observables \(O_A=O_A^\dagger \) and \(O_B=O_B^\dagger \), one has \(P O_A(\mathbb {1}-P)O_B P=0\). This is [ CPGSV16 , Definition D.1, lines 2187–2191 ] .

There are orthogonal projectors \(P_{AX}\) and \(P_{XB}\) on the two local factors whose lifts commute and whose ranges satisfy the source ground-space intersection condition \(\operatorname{ran}P=\operatorname{ran}\widehat P_{AX}\cap \operatorname{ran}\widehat P_{XB}\). Thus \(Q_{AX}=\mathbb {1}-P_{AX}\) and \(Q_{XB}=\mathbb {1}-P_{XB}\) are exactly the commuting parent terms of [ CPGSV16 , Definition D.2, lines 2205–2218 ] .

Let \(A\) be a matrix product tensor. Write its canonical two-site parent interaction in the ordered-pair basis of the two-site physical Hilbert space, and denote the resulting matrix by \(Q_A\).

Assume \(D{\gt}0\), let \(\rho \) be positive definite, and assume that \(\Gamma _{L+Q}^{\mathrm{ES}}\) and \(\Gamma _{K+L}^{\mathrm{ES}}\) are injective. The centered physical sandwich is obtained by placing \(R_{K,L,Q}\) between the reassociated tail pseudoinverse and the left boundary pseudoinverse.

Assume \(\operatorname {tr}(\rho )\neq 0\) and let \(\mathcal K_\infty =\mathscr R(P_\rho )\). Define

Assume \(D{\gt}0\), let \(\rho \) be positive definite, and assume that \(\Gamma _{L+Q}^{\mathrm{ES}}\) and \(\Gamma _{K+L+Q}^{\mathrm{ES}}\) are injective. The full correction is the physical sandwich containing \((\mathcal K_{K+L+Q})^{-1}-\mathcal K_\infty ^{-1}\).

Assume \(D{\gt}0\), let \(\rho \) be positive definite, and assume that \(\Gamma _{L+Q}^{\mathrm{ES}}\) and \(\Gamma _{K+L}^{\mathrm{ES}}\) are injective. The left correction is the physical sandwich containing \(\mathcal K_{K+L}-\mathcal K_\infty \) and the left boundary pseudoinverse.

For \(X=(X_u)_{u\in \Omega _K}\), define

For \(K,L,Q\in \mathbb {N}\), let

be the unitary coordinate map induced by associativity of addition.

For \(Z=(Z_j)_{j\in \Omega _Q}\), define

Assume \(D{\gt}0\), let \(\rho \) be positive definite, and assume that \(\Gamma _{L+Q}^{\mathrm{ES}}\) and \(\Gamma _{K+L+Q}^{\mathrm{ES}}\) are injective. The tail correction is the physical sandwich containing \(\mathcal K_{L+Q}-\mathcal K_\infty \) between the reassociated tail pseudoinverse, \(J_K^{\mathrm{tail}}\), and the full boundary pseudoinverse.

Let \(P_i\) be a finite family of symmetric projections and let \(H=\sum _iP_i\). Suppose the anticommutator forms obey

with row sums \(\sum _{j\ne i}c_{ij}\le 1\), column sums \(\sum _{i\ne j}c_{ij}\le 1\), and \(\gamma \le 1\). Then \(H^2\ge \gamma H\) as a quadratic form.

Let \(A^i=X\Lambda U^iX^{-1}\) be the Appendix B structural form, and write \(A_{\mathrm{core}}^i=\Lambda U^i\). The canonical local parent interaction is unchanged by this basis change: \(q_L(A)=q_L(A_{\mathrm{core}})\).

Let \(Q_{AX}\) and \(Q_{XB}\) be the two-site coefficient-space transports of source projectors satisfying the Definition D.2 local condition on \(A X B\). If these transported projectors agree with the Appendix B coefficient representatives,

then \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) satisfy the algebraic overlapping two-site condition of Definition 13.1.2. Consequently, \(h_0^{(3)}(A,2)h_1^{(3)}(A,2) = h_1^{(3)}(A,2)h_0^{(3)}(A,2)\).

If the Appendix B \(AX\) and \(XB\) coefficient representatives satisfy the lifted Definition D.2 commutator

then the corresponding first two translated length-two local terms on the three-site window commute: \(h_0^{(3)}(A,2)h_1^{(3)}(A,2) = h_1^{(3)}(A,2)h_0^{(3)}(A,2)\). The same conclusion holds if one supplies a Definition D.2-shaped local condition whose two operators are these coefficient-space representatives, rather than the source projectors themselves.

Suppose the lifted \(AX\) and \(XB\) representatives commute:

Then \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) satisfy the algebraic overlapping two-site condition of Definition 13.1.2.

If the Appendix B \(AX\) and \(XB\) coefficient representatives satisfy the overlapping two-site commutation datum

then the corresponding first two translated length-two local terms on the three-site window commute: \(h_0^{(3)}(A,2)h_1^{(3)}(A,2) = h_1^{(3)}(A,2)h_0^{(3)}(A,2)\).

The two Appendix B two-site coefficient representatives are idempotent:

As coefficient-space representatives, \(\widehat Q_{AX}=q_2(A)\) and \(\widehat Q_{XB}=q_2(A)\).

On the three-site \(A,X,B\) window, the two adjacent length-two local terms are the two local lifts

The Schmidt rank \(r_a\) of the nonzero loop bond vector is positive.

If \(a\) and \(b\) are distinct positive loops in the sector graph, then the mixed transfer map of their normalized minimal loop tensors vanishes:

Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), where \(\mu _j\ne 0\). Suppose that, for every \(N{\gt}L\), \(G_L^{(N)}(B) =\bigvee _{j=0}^{r-1}\operatorname{span}\{ V^{(N)}(A_j)\} \). Then the source parent-Hamiltonian spanning equation holds: \(\ker H_N(B,L) =\operatorname{span}\{ V^{(N)}(A_j):j=0,\ldots ,r-1\} \). It is enough to prove the two equations

Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\). Suppose \(N{\gt}0\), \(L\le N\), and the periodic chain constraints for \(B\) split into the block constraints:

If every block chain space is contained in its periodic MPS line,

then the zero-energy space of the parent Hamiltonian is contained in the BNT vector span:

Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\ne 0\). Suppose the physical alphabet and all block dimensions are nonempty. Assume that the block family is irreducible, left-canonical, has normalized self-overlap, and has no two blocks gauge-phase equivalent. Suppose also that each \(A_j\) is \(L_0\)-block-injective, that \(L_0{\gt}0\), and that \(\sum _a A^j_a(A^j_a)^\dagger =I \qquad (0\le j{\lt}r)\). Let \(N{\gt}0\), \(L{\gt}0\), \(L\le N\), \(L+L_0\le N\), and \((L_0+1) +(r-1)((L_0+1)+((L_0+1)+(L_0+1)))+1 \le L\). Assume that, for every boundary-crossing interval beginning at \(i\), the simultaneous products \(w\longmapsto (A^1_w,\ldots ,A^r_w), \qquad |w|=N-i\), span the product algebra \(\prod _jM_{D_j}(\mathbb {C})\). If every block chain space is contained in its periodic MPS line,

then

Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\ne 0\). Suppose the physical alphabet and all block dimensions are nonempty. Assume that the block family is irreducible, left-canonical, has normalized self-overlap, and has no two blocks gauge-phase equivalent. Suppose also that each \(A_j\) is \(L_0\)-block-injective, that \(L_0{\gt}0\), and that \(\sum _a A^j_a(A^j_a)^\dagger =I \qquad (0\le j{\lt}r)\). Let \(N{\gt}0\), \(L{\gt}0\), \(L\le N\), \(L+L_0\le N\), and \((L_0+1) +(r-1)((L_0+1)+((L_0+1)+(L_0+1)))+1 \le L\). Assume the opened-boundary \(C^j,D^j\) comparison: whenever \(\psi \in \mathcal G_{N,L}(B)\) and \(\psi =\Gamma _N^B(\bigoplus _j X_j)\), there are matrices \(C^j_{i,\rho }\) such that, for every boundary-crossing interval \(i\), every \(\rho \), and every \(\beta \), \(A^j_\beta C^j_{i,\rho } =((\mu _j^N X_j)A^j_\beta )A^j_\rho \). If every block chain space is contained in its periodic MPS line,

then

Let \(B=\bigoplus _{j=0}^{r-1}\mu _jA_j\), with every \(\mu _j\ne 0\). Then, for every \(N{\gt}L\), the source parent-Hamiltonian spanning condition \(\ker H_L^{(N)}(B) =\operatorname{span}\{ V^{(N)}(A_j):j=0,\ldots ,r-1\} \) is equivalent to the reverse containment, for the same \(N\):

In the nearest-neighbor case \(L=2\), the all-chain source condition of Definition 13.1.18 is equivalent to the conjunction

Assume \(p{\gt}0\) and \(N\geq 2\). Write \(h^{A,L}_j\) for the range-\(L\) local term of a tensor \(A\) at site \(j\). Let \(B=\operatorname {block}_p(A)\), and let \(U\) be the canonical configuration isometry from the blocked chain of length \(N\) to the original chain of length \(Np\). For every blocked site \(i\),

Hence the conjugated blocked Hamiltonian is the aligned sparse sum

and

in Loewner order.

Let \(A\) be an MPS tensor with nonzero physical dimension, let \(p\) be any blocking length, and set \(B=\operatorname {block}_p(A)\). Then

If \(B\) is injective, then the bound on either side

gives the adjacent three-site anticommutator inequality

The same quadratic-form inequality holds for every forward or backward adjacent pair of cyclic range-two terms on every chain of length at least three, including pairs crossing the periodic cut.

With the hypotheses and notation of Lemma 14.3.2.10, one has

This records the ground-space equality conclusion of the conditional crossing-span version above. It is the corresponding part of [ PGVWC07 , Theorem 12 ] , but only under the visible tail-word span hypothesis and in the longer stated range. The independence of the \(N\)-site spaces is a separate conclusion of Lemma 14.3.2.10. This intermediate statement remains conditional on the short tail-word spans. The source-range conclusion without them is supplied by the global change-of-cut theorem Theorem 14.3.1.41 under the unrestricted source normalization, with positive definite dual fixed points satisfying

The doubly normalized theorem with \(\Lambda _j=\mathbb {1}\) remains available as a specialization of that result.

With the hypotheses and notation of Lemma 14.3.2.8, one has

This records only the ground-space equality conclusion obtained after the \(C^j,D^j\) boundary-condition comparison input in [ PGVWC07 , Theorem 12, proof lines 1446–1456 ] ; the local normalized \(E^j\) calculation is Lemma 14.2.1.10. The independence of the \(N\)-site spaces is a separate conclusion of Lemma 14.3.2.8. The statement is still in the length-\(L_0\) injectivity range displayed in that lemma, not the shorter range in [ PGVWC07 , Theorem 12 ] .

For a family \(A_0,\ldots ,A_{r-1}\) and a chain length \(N\), let \(M_j=\operatorname{span}\{ V^{(N)}(A_j)\} \). Then

Let \(B\) be a canonical-form tensor with BNT components \(A_1,\ldots ,A_g\). If every component vector has zero energy for the parent Hamiltonian, \(H_L^{(N)}(B)V^{(N)}(A_j)=0\) for \(1\le j\le g\), then

It is enough to assume the local frustration-free equations \(h_iV^{(N)}(A_j)=0\) for every \(i\) and \(j\). For \(N{\gt}0\) and \(L\le N\), in the block-diagonal case \(B=\bigoplus _j\mu _jA_j\), with every \(\mu _j\ne 0\), the same inclusion follows because each local space \(G_L(A_j)\) is contained in \(G_L(B)\). This is the easy inclusion in the source ground-space spanning equation of [ CPGSV16 , Definition 3.9 ] ; the reverse inclusion is the remaining spanning direction.

Assume \(N{\gt}0\) and \(L\le N\). Then \(G_L^{(N)}(A)\subseteq \ker H_N(A,L)\).

Let \(L\le N\). Suppose that \(h_i h_j=h_j h_i\) whenever the length-\(L\) cyclic supports of the two local terms meet. Then the parent Hamiltonian has commuting local terms: \(h_i h_j=h_j h_i\) for all \(i,j\). In the case \(L=2\), this gives the nearest-neighbor commuting condition.

If \(P\) and \(Q\) are commuting idempotents, then \(Q\circ (\mathbb {1}-P\circ Q)\circ P=0\).

If \(L\le N\) and the cyclic supports \(S_i\) and \(S_j\) do not meet, then the Hilbert-space local terms have non-negative ordered cross term: \(0\le \operatorname{Re}\langle h_{i,\mathrm{ES}}v,h_{j,\mathrm{ES}}v\rangle \).

If \(2L\le N\) and \(L{\gt}1\), then, for every \(i\in \{ 0,\ldots ,N{-}1\} \), \(\# \{ j\ne i: i\sim _L j\} \le 2(L-1)\).

If \(L\le N\), then a site \(k\) belongs to the cyclic support \(S_i\) exactly when its cyclic offset from \(i\) is below \(L\): \(k\in S_i \quad \Longleftrightarrow \quad (k-i)\bmod N {\lt} L\).

For length-\(L\) cyclic windows, \(i\sim _Lj\) if and only if \(j\sim _Li\).

The Hilbert-space local projector \(P_L^{\mathrm{ES}}(A)\) is positive. Consequently, every summand \(R_{i,\tau }^{\dagger }P_L^{\mathrm{ES}}(A)R_{i,\tau }\) is positive, and so is the finite sum over \(\tau \).

For any words \(w_1, w_2\), \(A^{w_1 \cdot w_2} = A^{w_1} A^{w_2}\).

Let \(P_i\) be a finite family of symmetric projections and let \(\gamma \le 1\). Suppose an interaction predicate \(i\sim j\) has at most \(m\) off-diagonal neighbours in each row and in each column, where \(m{\gt}0\). If noninteracting pairs have non-negative anticommutator form and interacting pairs satisfy

then \(H=\sum _iP_i\) satisfies \(H^2\ge \gamma H\) as a quadratic form.

Let \(P_i\) be a finite family of symmetric projections and let \(\gamma \le 1\). Suppose an interaction predicate has at most \(m{\gt}0\) off-diagonal neighbours in each row, noninteracting ordered cross terms are non-negative, and interacting pairs satisfy the norm-compression estimate

Then \(H=\sum _iP_i\) satisfies \(H^2\ge \gamma H\) as a quadratic form. The same conclusion holds from a separate compression coefficient \(\eta \) whenever \(\eta \le (1-\gamma )/m\). In particular, if \(0\le \eta \) and \(\eta m{\lt}1\), then \(H^2\ge (1-\eta m)H\) and \(1-\eta m{\gt}0\).

Let \(P_i\) be a finite family of symmetric projections and let \(\gamma \le 1\). Suppose an interaction predicate \(i\sim j\) has at most \(m\) off-diagonal neighbours in each row for some positive integer \(m\). If noninteracting ordered cross terms are non-negative and, on each interacting pair,

then \(H=\sum _iP_i\) satisfies \(H^2\ge \gamma H\) as a quadratic form.

For any endomorphisms \(P\) and \(Q\),

For every boundary matrix \(X\in M_{D}(\mathbb {C})\) and every length-\(L\) word \(\sigma \), one has \(\Gamma _L(X)(\sigma )=\operatorname{tr}(A^\sigma X)\).

The Hilbert-space parent Hamiltonian is the sum of the Hilbert-space local terms, \(H_{N,\mathrm{ES}}=\sum _i h_{i,\mathrm{ES}}\), and each local term admits the exact cyclic-averaging formula

For \(L\le N\), a Hilbert-space local term \(h_{i,\mathrm{ES}}\) annihilates a vector \(v\) if and only if every restriction of \(v\) to the cyclic length-\(L\) window starting at \(i\), with the outside configuration fixed, lies in \(G_L^{\mathrm{ES}}(A)\).

For every site \(i\), one has \(h_i(A,L)^2=h_i(A,L)\).

For \(v\in \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\), one has \(v\in G_L^{\mathrm{ES}}(A)\) if and only if \(\iota _L^{-1}v\in G_L(A)\).

A vector belongs to \(\mathcal G_{N,L}^{\mathrm{ES}}(A)\) exactly when the parent Hamiltonian annihilates it.

Let \(A=(A_i)_i\) and \(B=(B_i)_i\) be two tensors, possibly with different bond dimensions, and let \(u\) be unitary on the physical index. If

then

Let \(U,A\) and \(V,B\) be pairs of tensors with respective common bond dimensions. Suppose that \(\zeta _A,\zeta _B\ne 0\) and that \(X_A,X_B\) are invertible matrices such that, for every physical index \(i\),

Then

Identifying equal bond dimensions on either tensor leg preserves this vanishing condition in both directions.

For a length-\(N\) physical configuration \(s=(s_0,\ldots ,s_{N-1})\), physical rotation satisfies

Let \(N\ge 2\). Suppose that, for every \(i\in \mathbb {Z}/N\mathbb {Z}\), the translated length-two local term commutes with its cyclic right neighbor:

Then all translated length-two local terms commute, and the family \(p_i=h_i(A,2)\) satisfies the finite-chain local-projector hypotheses used in the conditional Appendix B extraction.

For every \(A\), \(L\), and \(N\), the operator \(H^{\mathrm{open}}_{N,L}(A)\) is positive.

Let \(P_i\) be a finite family of symmetric projections and let \(H=\sum _iP_i\). If the ordered off-diagonal forms obey

with row sums \(\sum _{j\ne i}c_{ij}\le 1\) and \(\gamma \le 1\), then \(H^2\ge \gamma H\) as a quadratic form.

Assume uniformly for every \(N\ge 2L\) that the Hilbert-space local terms satisfy the anticommutator martingale estimate

for \(i\ne j\), with row and column coefficient sums bounded by one. If \(L{\gt}1\), then \(1/(4L){\gt}0\) and, for every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\frac{1}{4L}\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).

Fix \(L{\gt}1\) and let \(i\sim _L j\) mean that the cyclic supports of the length-\(L\) windows at \(i\) and \(j\) meet. Suppose uniformly for every \(N\ge 2L\) that overlapping off-diagonal terms satisfy

Then the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).

Fix \(L{\gt}1\), and let \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) and every overlapping off-diagonal pair that

If \(\gamma {\gt}0\), \(\gamma \le 1\), and \(\eta \le (1-\gamma )/m\), then \(0{\lt}\gamma \) and, for every admissible \(N\) and every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). In particular, if \(\eta \ge 0\) and \(\eta m{\lt}1\), then \(0{\lt}1-\eta m\) and, for every admissible \(N\) and every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \((1-\eta m)\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). This strict all-vector compression form is a stronger conditional sufficient condition, not the source principal-angle estimate itself. It is a sufficient strengthening of the anticommutator martingale estimate in Cirac–Perez-Garcia–Schuch–Verstraete  [ CPGSV21 , Section IV.C ] . It does not prove the remaining MPS-specific estimate; it isolates the finite-row martingale argument once such a strengthening is supplied.

Fix \(L{\gt}1\). Suppose uniformly for every \(N\ge 2L\) and every overlapping off-diagonal pair that

Then the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).

Fix \(L{\gt}1\), and let \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) and every overlapping off-diagonal pair that

that is, a bound \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| \le \eta \) on the operator-product norm. If \(\gamma {\gt}0\), \(\gamma \le 1\), and \(\eta \le (1-\gamma )/m\), then \(0{\lt}\gamma \) and, for every admissible \(N\) and every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). In particular, if \(\eta \ge 0\) and \(\eta m{\lt}1\), then \(0{\lt}1-\eta m\) and the same norm bound holds with \(\gamma =1-\eta m\).

This is the symmetric (operator-product) counterpart of Lemma 13.3.2.12, obtained from the anticommutator conversion recorded in Remark 13.3.1.8. Like that lemma it is a conditional reduction: the excitation projections satisfy \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| =1\) whenever the overlapping ranges share a common nonzero vector, so the hypothesis is unsatisfiable for \(\eta {\lt}1\) in that case. It isolates the finite-row martingale argument once such a bound is supplied; the gap with the cited principal-angle estimates is documented in the paper-gap note [ con26m ] .

Fix \(L{\gt}1\) and let \(i\sim _L j\) mean that the cyclic supports of the length-\(L\) windows at \(i\) and \(j\) meet. Suppose uniformly for every \(N\ge 2L\) that overlapping off-diagonal terms satisfy

Then the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).

Fix \(L{\gt}1\) and let \(i\sim _L j\) mean that the cyclic supports of the length-\(L\) windows at \(i\) and \(j\) meet. Suppose uniformly for every \(N\ge 2L\) that overlapping off-diagonal terms satisfy

Then \(1/(4L){\gt}0\), and the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).

Let \(H_{N,L}^{\mathrm{ES}}(A)\) be the Hilbert-space representative of the parent Hamiltonian. If a constant \(\gamma {\gt}0\) satisfies the uniform quadratic-form estimate

for every \(N\ge 2L\) and every vector \(v\), then \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \) for every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \). In particular, with \(L{\gt}1\) and \(\gamma =1/(4L)\), the explicit gap-bound conclusion follows once this same quadratic-form estimate (30) is supplied.

The periodic MPS vector is a frustration-free ground state: for every site \(i\), one has \(h_iV^{(N)}(A)=0\).

Fix \(L{\gt}1\) and set \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) that the overlap relation on local windows has at most \(m\) off-diagonal neighbours in each row and in each column. If non-overlapping local terms have non-negative anticommutator form, and overlapping local terms satisfy

then the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).

For a fixed chain length \(N\), let \(m{\gt}0\) and \(\gamma \le 1\). Suppose an overlap predicate on local windows has at most \(m\) overlapping off-diagonal windows in each row, non-overlapping local terms have non-negative ordered cross terms, and overlapping local terms satisfy

Then \(H_{N,L}^{\mathrm{ES}}{}^2\ge \gamma H_{N,L}^{\mathrm{ES}}\) as a quadratic form. The same conclusion holds from a separate coefficient \(\eta \) satisfying \(\eta \le (1-\gamma )/m\).

Fix \(L{\gt}1\) and set \(m=2(L-1)\). Suppose uniformly for every \(N\ge 2L\) that the hypotheses of the fixed-chain finite-overlap martingale reduction hold with \(\gamma =1/(4L)\) and this value of \(m\). Then \(1/(4L){\gt}0\), and the explicit norm lower bound with constant \(1/(4L)\) follows for vectors in \((\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \).

For a fixed chain length \(N\), let \(m\) be a positive integer and let \(\gamma \le 1\). Suppose an overlap predicate on local windows has at most \(m\) overlapping off-diagonal windows in each row. If non-overlapping local terms have non-negative ordered cross terms, and overlapping local terms satisfy the one-sided ordered cross-term estimate

and the Hilbert-space local terms are symmetric projections, then the Hilbert-space Hamiltonian satisfies \(H_{N,L}^{\mathrm{ES}}{}^2\ge \gamma H_{N,L}^{\mathrm{ES}}\) as a quadratic form.

Assume uniformly for every \(N\ge 2L\) that the Hilbert-space local terms satisfy the ordered row-sum estimate (38) with \(\gamma =1/(4L)\). If \(L{\gt}1\), then \(1/(4L){\gt}0\) and, for every \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\frac{1}{4L}\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).

For a fixed chain length \(N\), suppose the Hilbert-space local terms \(h_{i,\mathrm{ES}}\) satisfy the ordered row-sum estimate

with \(\sum _{j\ne i}c_{ij}\le 1\) and \(\gamma \le 1\). Then the Hilbert-space parent Hamiltonian satisfies

for every vector \(v\).

The Hilbert-space local interaction \(P_L^{\mathrm{ES}}(A)\) annihilates exactly the Hilbert-space local ground space: \(P_L^{\mathrm{ES}}(A)v=0\) if and only if \(v\in G_L^{\mathrm{ES}}(A)\).

The local parent interaction is a projector: \(h_L(A)^2=h_L(A)\).

For finite-dimensional spaces \(H_L\) and \(H_R\), one has \(\operatorname{tr}_R(\mathbb {1}_{L\otimes R})=(\dim H_R)\mathbb {1}_L\).

Let \(P\) and \(Q\) be finite-dimensional complex endomorphisms, and let \(\gamma \geq 0\). Suppose that \(P\) is positive and

for every \(v\in (\ker P)^\perp \). Then

on \((\ker P)^\perp \). If also \(P\leq Q\) and \(\ker P=\ker Q\), then

for every \(v\in (\ker Q)^\perp \).

For every boundary matrix \(X\in M_{D}(\mathbb {C})\) and every suffix \(\tau \),

For \(L=1\), the prefix restriction \(P_\tau \) coincides with the last-site restriction: \(P_\tau \psi =R^{\mathrm{last}}(\psi ,\tau (0))\) for every \(\tau \colon \{ 0,\ldots ,1{-}1\} \to \{ 0,\ldots ,d{-}1\} \).

Let \(N\ge 2\). Suppose that the translated length-two local terms commute for every pair of overlapping cyclic windows: \(h_i(A,2)h_j(A,2) = h_j(A,2)h_i(A,2)\). Then the family \(p_i=h_i(A,2)\) gives the finite-chain local-projector hypotheses used in the conditional Appendix B extraction.

Let \(H\) be a positive semidefinite self-adjoint operator on a finite-dimensional complex Hilbert space, and suppose that, for all \(v\), \(\gamma \langle v,Hv\rangle \le \langle Hv,Hv\rangle \), i.e., \(H^2\ge \gamma H\) as a quadratic form, for some \(\gamma {\gt}0\). Then, for every \(v\) in the orthogonal complement of \(\ker H\), \(\gamma \| v\| \le \| Hv\| \). In particular, every positive eigenvalue of \(H\) is at least \(\gamma \). The positive-semidefiniteness hypothesis is essential: without it, \(H=-\mathbb {1}\) with \(\gamma =2\) satisfies the quadratic-form inequality vacuously but violates the norm bound.

A virtual similarity preserves the physical blocking relation and hence transfer idempotence. It also preserves every local MPS space, so the canonical parent interaction and all its translated local terms are unchanged.

The Hilbert-space local projector \(P_L^{\mathrm{ES}}(A)\) is a symmetric projection, and every Hilbert-space local term \(h_{i,\mathrm{ES}}\) on the \(N\)-site chain is a symmetric projection.

Let \(L\le N\) and let \(h_{i,\mathrm{ES}},h_{j,\mathrm{ES}}\) be Hilbert-space local terms whose cyclic \(L\)-windows are site-disjoint: no site of the periodic chain has cyclic offset \({\lt}L\) from both \(i\) and \(j\). Then the terms commute pointwise: for every vector \(v\), \(h_{i,\mathrm{ES}}(h_{j,\mathrm{ES}}v) =h_{j,\mathrm{ES}}(h_{i,\mathrm{ES}}v)\). The same commutation identity holds for the corresponding local terms before passing to the Hilbert-space model. Their ordered cross term is non-negative: \(0\le \operatorname{Re}\langle h_{i,\mathrm{ES}}v,h_{j,\mathrm{ES}}v\rangle \).

Each Hilbert-space local term \(h_{i,\mathrm{ES}}\) is positive, hence the full Hilbert-space parent Hamiltonian \(H_{N,\mathrm{ES}}\) is positive.

The lifts from \(AX\) and \(XB\) preserve multiplication,

and each of the four lifts from \(AX\), \(XB\), \(A\), and \(B\) preserves adjoints:

Let \(P_U\) be the orthogonal projection onto \(U\). If the Friedrichs overlap of \(U\) and \(V\) is at most \(\eta \), where \(\eta \ge 0\), then

for every \(v\in V\cap (U\cap V)^\perp \).

Under the hypotheses of Lemma 13.3.1.5, every \(v\in U\cap (U\cap V)^\perp \) satisfies

Let \(P_U\) and \(P_V\) be the orthogonal projections onto \(U\) and \(V\). If the Friedrichs overlap of \(U\) and \(V\) is at most \(\eta \), where \(\eta \ge 0\), then every \(v\in (U\cap V)^\perp \) satisfies

Let \(P\) and \(Q\) be the orthogonal projections onto \(K^\perp \) and \(L^\perp \). If the Friedrichs overlap of these two ranges is at most \(\eta \ge 0\), then every \(v\in K^\perp \cap (K^\perp \cap L^\perp )^\perp \) satisfies

For \(i,j\in \mathbb {Z}/N\mathbb {Z}\), if the two length-two cyclic supports starting at \(i\) and \(j\) overlap, then

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

The listed identities are the equations for \(s=0\), \(M=N\), for deleting the first or last site, and for first fixing \(K\) sites and then 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 \(\rho ^{\mathrm{cyc}}_{i,L,\omega }(i+r)=\omega (r)\). Here site labels are taken modulo \(N\). The remaining listed identities say that deleting the final site of an \((L+1)\)-interval gives the \(L\)-interval beginning at \(i\); if \(L+1\le N\), deleting the first site gives the \(L\)-interval beginning at \(i+1\); and the cyclic formula agrees with the contiguous formula when the interval does not wrap around the ring.

Let \(P_L^{\mathrm{ES}}(A)\) denote the orthogonal projector onto \((G_L^{\mathrm{ES}}(A))^\perp \) on the canonical \(\ell ^2\) \(L\)-site Hilbert space. For a cyclic window starting at \(i\) and an outside configuration \(\tau \), the map \(R_{i,\tau }\) restricts an \(N\)-site vector to that length-\(L\) cyclic window with the outside configuration fixed by \(\tau \). The local term \(h_{i,\mathrm{ES}}\) is the corresponding \(N\)-site local projector, and \(R_{i,\tau }^{\dagger }P_L^{\mathrm{ES}}(A)R_{i,\tau }\) is the basic positive summand in its cyclic-averaging formula.

The nearest-neighbor condition is the length-\(2\) specialization of the general commuting condition. The ground-vector condition above contains this commuting equation together with the frustration-free equation for \(V^{(N)}(A)\); it does not assert the source ground-space spanning property. For \(N\ge 2\), Lemma 12.3.63 with \(L=2\) gives \(h_iV^{(N)}(A)=0\) for every site \(i\). The fixed-chain ground-space condition also contains \(\ker H_2^{(N)}(B) = \operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \), and quantifying this condition over all \(N{\gt}2\) gives the nearest-neighbor case of the spanning condition above. Conversely, the all-chain condition is exactly the conjunction of the all-chain ground-vector equations and the nearest-neighbor instance of the spanning condition above. This is precisely the condition named in Definition 13.1.18. Since the displayed zero-energy equation follows from the parent-Hamiltonian construction for \(N{\gt}2\), independently of the commutation and spanning equations, this is equivalently

The Appendix B structural form supplies \(A^i=X\Lambda U^iX^{-1}\). In the source unit pair-index convention, the residual tensor satisfies

Therefore the residual core tensor \(\Lambda U^i\) satisfies the weighted pair-index relation

This is an orthogonality relation after summing over the physical letter \(i\). It is not a coefficient identity for a fixed physical word \(\sigma \). The coefficient associated to the source basic-vector formula is

This is the coefficient expression for the source vector

where \(\varphi _j\) is shared by \(b_t\) and \(a_{t+1}\) and \(U\) acts on \((a_t,b_t)\). At length two, this cyclic coefficient is the structural two-site amplitude: \(C_2(\sigma )=\phi _2(\sigma )\). The change of basis does not alter either the local spaces \(G_L\) or the periodic MPS coefficients: \(G_L(A)=G_L(\Lambda U)\). It therefore also leaves the canonical parent interaction unchanged: \(q_L(A)=q_L(\Lambda U)\). At length two, the Appendix B two-site basic support is \(G_2(\Lambda U)\), and the coefficient-space representatives \(\widehat Q_{AX}\) and \(\widehat Q_{XB}\) are the two copies of \(q_2(\Lambda U)\) before they are lifted to the \(AX\) and \(XB\) faces. Hence

as coefficient-space representatives. On the three-site window their two lifts give the local identifications

Consequently, for every chain length \(L\geq 1\), the original tensor has the same cyclic coefficient formula: \(V^{(L)}(A)_\sigma =C_L(\sigma )\). In particular, the corresponding two-site amplitude is

Thus the one-pair case of the factorization is already verified for the core tensor: \(V^{(2)}(\Lambda U)(\sigma )=\phi _2(\sigma _0,\sigma _1)\). The even-chain physical-pair factorization is the following condition, for a positive number \(n\) of pairs,

The case \(n=0\) is not part of this condition: the empty-chain MPV coefficient is the virtual bond dimension, whereas the empty product on the right-hand side is \(1\). The first equality, \(V^{(L)}(\Lambda U)_\sigma =C_L(\sigma )\), is the trace expansion of the closed matrix product and is the coefficient form of \(U^{\otimes L}\varphi _j^{\otimes L}\). The remaining coefficient step is to relate this source expression, with the physical word \(\sigma \) fixed, to the even-chain physical-pair factorization above. On a three-site chain, the two adjacent length-two local terms are the \(AX\) and \(XB\) actions of the canonical two-site parent interaction. Theorem 13.1.57 constructs the adjacent physical support projectors from the basic-vector expression and proves the overlapping relation

Theorem 13.1.60 transports this identity around a periodic chain. Consequently, for every \(N{\gt}2\) and every pair of sites \(i,j\in \mathbb {Z}/N\mathbb {Z}\), \(h_i(A,2)h_j(A,2) = h_j(A,2)h_i(A,2)\). Definition 13.1.62 collects these maps into a commuting idempotent family; idempotency is Lemma 12.3.57. Together with Lemma 12.3.63, this gives the zero-energy equation for \(V^{(N)}(A)\). Neither conclusion requires the even-chain physical-pair factorization displayed above. That factorization remains a separate conditional coefficient statement. The full source implication additionally requires removal of the left-canonical hypothesis and the ground-space spanning equation. The coefficient-space analog of the local kernel-intersection equation is Theorem 13.1.59; identifying its hatted maps with the source projectors remains separate and is not needed for the chain commutator.

The projection-geometry reduction uses the following elementary Hilbert-space identities: the diagonal identity \(\operatorname{Re}\langle Pv,Pv\rangle =\operatorname{Re}\langle Pv,v\rangle \) for symmetric projections; nonnegativity of this quadratic form; nonnegativity of \(\operatorname{Re}\langle Pv,Qv\rangle \) for commuting symmetric projections \(P,Q\); the Cauchy–Schwarz conversion from \(\| PQv\| \le c\| Pv\| \) to \(\operatorname{Re}\langle Pv,Qv\rangle \ge -c\operatorname{Re}\langle Pv,v\rangle \); the symmetric variant which, for \(c\ge 0\), converts the operator-product bound \(\| PQv\| \le c\| Qv\| \) to the anticommutator estimate \(\operatorname{Re}\langle Pv,Qv\rangle +\operatorname{Re}\langle Qv,Pv\rangle \ge -c(\operatorname{Re}\langle Pv,v\rangle +\operatorname{Re}\langle Qv,v\rangle )\); finite-sum expansions of \(\operatorname{Re}\langle (\sum _iP_i)v,v\rangle \) and \(\operatorname{Re}\langle (\sum _iP_i)v,(\sum _iP_i)v\rangle \); the conclusion that a vector annihilated by a finite sum of symmetric projections is annihilated by each projection; the diagonal/off-diagonal split of the ordered double sum; the aggregate off-diagonal estimate obtained from row sums \(\sum _{j\ne i}c_{ij}\le 1\); and the corresponding aggregate estimate for the anticommutator martingale condition.

If \(A\) is injective and \(L\ge 2\), then on an \((L+1)\)-site chain the kernels of the two adjacent Hilbert-space \(L\)-site local terms have common kernel equal to the Hilbert-space \((L+1)\)-site MPS ground space:

Let \(\iota _L\) denote the canonical identification of the \(L\)-site state space with its Hilbert-space realization, and let \(R^{\mathrm{cyc}}_{i,\tau }\) denote cyclic restriction to the length-\(L\) window beginning at \(i\), with background configuration \(\tau \). For every \(L\in \mathbb {N}\), every \((L+1)\)-site Hilbert-space vector \(v\), and every background configuration \(\tau \),

On three sites, for every one-site prefix \(u\) and every state \(\psi \),

Finally, if \(L{\gt}0\), then for every \((L+1)\)-site Hilbert-space vector \(v\) and every background configuration \(\tau \),

No hypothesis on the physical or virtual dimension is required.

For a tensor \(B\), let \(h_0\) and \(h_1\) be the two translated length-two parent interactions on the three-site Hilbert space. Then

Consequently, the orthogonal projectors onto the complementary excitation subspaces satisfy

These identities require neither injectivity nor a nonzero bond dimension.

Let \(L{\gt}1\). Suppose that for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,

in quadratic form. Then the parent Hamiltonian has the explicit gap constant \(\gamma _L:=1/(4L){\gt}0\). Concretely, if \(N\ge 2L\) and \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), then \(\gamma _L\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).

For every Appendix B structural datum and every periodic chain of length \(N{\gt}2\), the adjacent translated length-two parent interactions commute.

Let \(A\) be injective and let an Appendix B structural datum be fixed. Put

with the two copies of \(q_2(A)\) lifted to the indicated faces of the three-site coefficient space. Then

These are the coefficient-space analogs of the idempotence, commutation, and kernel-intersection equations in [ CPGSV16 , Definition D.2 ] . The hatted operators are not identified here with the source projectors on \(\mathcal H_A\otimes \mathcal H_X\) and \(\mathcal H_X\otimes \mathcal H_B\). For a normal left-canonical RFP tensor, injectivity follows from the RFP structural theorem and is not an additional hypothesis.

The two Appendix B coefficient representatives satisfy

and their adjacent actions on the common three-site coefficient space commute:

Thus they satisfy the algebraic overlapping two-site condition of Definition 13.1.2. This is the coefficient-space analog of the projector-and-commutator part of [ CPGSV16 , Definition D.2 ] . The hatted maps have not yet been identified with the source projectors on the explicitly factored spaces. By itself the result does not prove the kernel-intersection equation in that definition and makes no assertion for every chain length. The coefficient-space kernel equation is established in the next two results.

Let \(A\) be equipped with Appendix B structural data \((X,\Lambda ,U)\) as in Definition 13.1.35. Its pair-index isometry equation

implies \(U^*U=1\). In particular, \(U\) is injective.

For every \(N\geq 0\), one has \(U^{*\otimes N}U^{\otimes N}=1\). Hence \(U^{\otimes N}\) is injective.

Suppose the conditional even-chain hypotheses give commuting two-site parent terms for a tensor \(B\). If the nearest-neighbor parent Hamiltonian also satisfies the ground-space spanning equation of Definition 13.1.16, with BNT components \(A_1,\ldots ,A_g\), then \(B\) satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition of Definition 13.1.18.

In particular, if a normal left-canonical RFP tensor has nonzero bond dimension, the same conditional Appendix B hypotheses, and the same ground-space spanning equation, then it satisfies the full all-chain condition in [ CPGSV16 , Theorem 3.10(iii) ] .

On the homogeneous three-single-site coefficient space \((\mathbb {C}^d)^{\otimes 3}\), the hatted Appendix B (AX) representative acts as \([\widehat Q_{AX}]_{AX} =[\widehat Q_{AX}]\otimes I_d\). This equation gives the coefficient-factor specialization of the left local action in [ CPGSV16 , Definition D.2 ] . It does not construct a projector for an arbitrary factorization \(\mathcal H_A\otimes \mathcal H_X\otimes \mathcal H_B\).

On the homogeneous three-single-site coefficient space \((\mathbb {C}^d)^{\otimes 3}\), the hatted Appendix B (XB) representative acts as \([\widehat Q_{XB}]_{XB} =I_d\otimes [\widehat Q_{XB}]\). This equation gives the coefficient-factor specialization of the right local action in [ CPGSV16 , Definition D.2 ] . It does not construct a projector for an arbitrary factorization \(\mathcal H_A\otimes \mathcal H_X\otimes \mathcal H_B\).

Opening the outer virtual bond in the two-site contraction of [ CPGSV16 , Equations (3.17)–(3.18) ] gives the following local-support family. For every boundary matrix \(Y\), \(U^{\otimes 2}I_\varphi (v_Y) = \Gamma _2(\Lambda U)(Y)\). Consequently, \(\operatorname{ran}(U^{\otimes 2}I_\varphi )=G_2(\Lambda U)\).

The projector \(P_2\) satisfies

In particular, \(q_2(\Lambda U)\) annihilates the image of \(U^{\otimes 2}I_\varphi \), while \(P_2\) fixes that image pointwise.

Let \((P_2)_{01}\) and \((P_2)_{12}\) be the two adjacent placements of the two-site basic support projector on three physical sites. Then \((P_2)_{01}(P_2)_{12} =(P_2)_{12}(P_2)_{01}\). This is the local three-site consequence of the parent-space construction and the basic-vector form in [ CPGSV16 , Definition 3.9, Equations (3.16)–(3.18), and Appendix B ] . It does not assert commutation for every chain length.

The two adjacent placements commute: \(P_{01}P_{12}=P_{12}P_{01}\). The following support-location schematic marks the two bonds on which the projectors act; it does not depict the doubled operator network for the equality above.

Strict positivity of the \(\lambda _a\) makes \(Y\mapsto v_Y\) surjective.

Suppose that \(A\) is equipped with Beigi sector data, with one-site spatial unitary \(U\). For \(N\geq 2\), let \(Q_N\) be the product of the translated complementary parent interactions and let \(\widetilde Q_N\) be the corresponding product in sector coordinates. Then \((U^*)^{\otimes N}Q_NU^{\otimes N} = \widetilde Q_N\).

Suppose that \(A\) is equipped with Beigi sector data. For \(N\geq 2\), let \(h_0,\ldots ,h_{N-1}\) be its translated length-two parent interactions and set \(Q_N=\prod _{n=0}^{N-1}(\mathbb {1}-h_n)\). Every vector fixed by \(Q_N\) belongs to the kernel of the periodic parent Hamiltonian \(H_N=\sum _n h_n\).

Let \(G\) be a finite directed graph whose edge weights are non-negative integers, and let

be the weighted sum over ordered cyclic vertex sequences. Suppose that \(s_N\) is constant for all sufficiently large \(N\). Then every edge that belongs to a directed cycle has weight one, every directed cycle is a repetition of a single loop, and, for every \(N{\gt}0\), \(s_N=\# \{ a:w(a,a){\gt}0\} \). This is the comparison in [ Bei12 , Section IV ] .

Let \(a\to a\) be a positive loop. For every \(N\geq 2\), its physical cyclic product-of-pairs state belongs to the kernel of the length-two periodic parent Hamiltonian. Equivalently, its Euclidean representative belongs to the parent ground space. This is only a membership assertion; it does not assert that the loop states span the ground space.

Suppose that the parent-ground-space dimension is eventually constant. For every \(N{\gt}2\), the positive-loop product states form a nonzero pairwise orthogonal family and

In particular, the same conclusion holds under the BNT and all-chain nearest-neighbor parent-ground-space hypotheses.

For every \(N{\gt}0\), the periodic matrix product vector of \(C_a\) has coefficient \(\prod _{n=0}^{N-1}\varphi _a(r_n,l_{n+1})\) on a cyclic configuration in sector \(a\), and coefficient zero on every other sector configuration. Consequently, the periodic vector of the physical loop tensor is exactly the sitewise-unitary image of this cyclic product-of-pairs state. The same identities hold for the tensor on the minimal Schmidt support. For the normalized minimal tensor it is \(\nu _a^{-N}\Psi _a^{(N)}\).

The tensor \(\widetilde C_a\) on the Schmidt support is injective, as is its image under the one-site unitary. In particular, the physical tensor is normal.

The transfer map of the unnormalized coordinate tensor is

Hence the transfer map of \(\widehat C_a\) is idempotent. This remains true after the physical unitary rotation. The normalized physical tensor is injective and therefore normal.

For every \(N\geq 2\), the periodic ground-space dimension is

where the sum is over ordered cycles and the indices are read modulo \(N\). This is equations (3)–(4) of [ Bei12 , Section III ] for chains of length at least two. The one-site periodic interaction requires a separate convention.

At period two, the ground-space dimension is

The sum is over ordered pairs: if \(a\ne b\), then \((a,b)\) and \((b,a)\) are distinct summands. This is the length-two convention stated after equation (4) of [ Bei12 , Section III ] .

Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\), and let \(A=\bigoplus _j A_j\) be their direct sum with one unit-weight copy of each sector. Then \((A_j)_j\) is a basis of normal tensors for \(A\): every \(A_j\) is a normal tensor, at every positive system length \(N\)

and the states \(\{ |V^{(N)}(A_j)\rangle \} _j\) are linearly independent for all sufficiently large \(N\).

Let \(S\) be a finite spectator index set, let \(V\) be the Euclidean virtual matrix space, and let \(G:V\to V\) be linear. If \(G^{\oplus S}\) acts on a boundary family \(x=(x_s)_{s\in S}\) by \((G^{\oplus S}x)_s=Gx_s\), then

This estimate also holds when \(S\) is empty.

Let \(S\) be a finite set and let \(G,H\colon V\to V\) be continuous linear endomorphisms of the virtual Euclidean space. Then

For every \(K\in \mathbb {N}\), the common overlap factors have fiber formulas

and norm bounds

The formulas and bounds remain valid for empty direct sums and incur no factor depending on the number of configurations.

Assume \(D\geq 1\) and that \(\rho \in M_{D}(\mathbb {C})\) is positive definite. The centered residual of Definition 13.3.5.44 factors exactly at the generally nonidentity limiting metric as

and hence

If, in addition, \(A\) is primitive with fixed-point witness \(\rho \), then there are constants \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that, for every \(K\in \mathbb {N}\) and every \(l\geq 1\),

All formulas in this theorem are reconstructed algebra. They are not sourced formulas and assert neither a full projector-defect estimate nor the FNW–Nachtergaele rational coefficient.

Whenever the three finite Grams are invertible, the common-ambient defect is

and

Under the hypotheses of Definition 13.3.5.52,

Under the hypotheses of Definition 13.3.5.52, the three corrections have the exact factorizations

Let \(\rho \in M_{D}(\mathbb {C})\) be positive definite and set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). Write \(J_{\rm t}\) and \(J_{\rm l}\) for the tail and left virtual maps, and write \(S_n=\mathcal K_n^{-1}\) whenever the length-\(n\) boundary map is injective. The virtual residual in the exact tail-after-left C3 projector factorization has the telescoping decomposition

where \(M_l=(\Gamma ^{\rm tail}_{K,l+1})^\dagger \Gamma ^{\rm left}_{K+l,1}\) and the superscript \(\oplus \) denotes the corresponding fiberwise spectator operator. Substituting this equality into the exact inverse-Gram factorization gives the same decomposition of the physical projector residual, sandwiched between the tail boundary map, the two local fiberwise inverse Grams, and the adjoint left boundary map.

The limiting product in the first parenthesis has, without an additional hypothesis, the explicit limiting mixed pairing

Consequently, pairing the first parenthesis against boundary families is exactly the same finite sum with \(\mathcal K_l-\mathcal K_\infty \) in place of \(\mathcal K_l\).

These identities are reconstructed operator algebra. They assert neither the numerical FNW/Nachtergaele contraction nor any identification of a physical ground projector with \(P_\rho \).

Fix \(K,L_0\in \mathbb N\). Suppose that the tail boundary map for \((K,L_0+1)\), the left boundary map for \((K+L_0,1)\), and \(\Gamma _{K+L_0+1}\) are injective. The difference between the product of the first two inverse-Gram range projectors and the third inverse-Gram range projector is exactly the common-factorization residual obtained from the tail and left virtual maps. Their ranges are, respectively, \(\mathcal V_{K,L_0+1}(A)\), \(\mathcal U_{K+L_0}(A)\), and \(\mathcal G_{K+L_0+1}(A)\). This identity asserts no numerical contraction estimate.

If \(A\) is injective, \(D \ge 1\), \(N \ge 2\), and \(1 {\lt} L \le N\), then \(\mathcal G_{N,L}(A) = \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).

Assume \(N{\gt}0\) and \(L\le N\). Let \(\psi \) be a vector in the periodic chain ground space \(G_L^{(N)}(A)\). Then \(\psi \) satisfies all local zero-energy equations: for every site \(i\), one has \(h_i(A,L)\psi =0\).

If \(P_{AX}\) and \(P_{XB}\) are idempotent endomorphisms satisfying \(P_{AX}\circ P_{XB}=P_K\) and \(P_{AX}\circ P_{XB}=P_{XB}\circ P_{AX}\), and \(A\)-observables commute with \(P_{XB}\) while \(B\)-observables commute with \(P_{AX}\), then regions \(A\) and \(B\) are decorrelated w.r.t. \(K\). This is the idempotent-product form of the backward direction of [ CPGSV16 , Appendix D, Section D.2, Proposition D.3 ] .

Let \(B\geq 0\) be a two-site operator whose overlapping placements commute: \((B\otimes \mathbb {1})(\mathbb {1}\otimes B) = (\mathbb {1}\otimes B)(B\otimes \mathbb {1})\). There are positive integers \(d_{q,l},d_{q,r}\), a one-site unitary \(U\), and positive semidefinite operators \(\eta _{q,h}\) on \(\mathcal H_{q,r}\otimes \mathcal H_{h,l}\) such that

This is the symmetric two-sided form obtained from [ Bei12 , Lemma 2.1 and Section III, equation (2) ] .

If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data and \(Q_{AX}=\mathbb {1}-P_{AX}\) and \(Q_{XB}=\mathbb {1}-P_{XB}\), then \(Q_{AX}\circ Q_{XB}=Q_{XB}\circ Q_{AX}\).

Assume \(s+L\leq N\). For every boundary matrix \(X\) and complementary configuration \(\tau \),

Every normal tensor in the sense of [ CPGSV16 , Section 2.2 ] is injective after blocking by some finite length; its positive bond dimension follows from normality.

Let \(A\) and \(B\) be normal tensors of positive, possibly different, bond dimensions. Suppose that, for some nonzero \(\zeta \in \mathbb {C}\), \(|V^{(N)}(B)\rangle =\zeta ^N|V^{(N)}(A)\rangle \) for \(N\geq 0\). Then the bond dimensions are equal and there is an invertible matrix \(X\) and a scalar \(\eta \) of unit modulus such that \(B^i=\eta X A^iX^{-1}\) for every physical letter \(i\). This is the gauge recovery used in [ CPGSV16 , Appendix A ] .

Let \(A\) be a normal tensor in the sense of [ CPGSV16 , Section 2.2 ] . Its bond dimension is necessarily positive, and there is a positive definite matrix \(\sigma \) such that the gauge obtained from \(\sigma \) is trace-preserving, primitive, and irreducible. No scalar rescaling is needed.

For \(Q_q\) as in Theorem 13.3.5.65,

Suppose that \(h_i\) and \(h_{i+d}\) commute whenever \(R\leq d\leq N-R\). Then the cross term is nonnegative for these offsets. If \(1\leq m\), \(R\leq m\), and \(2m\leq N\), it follows that

Let \(h_i\), with \(i\in \mathbb Z/N\mathbb Z\), be orthogonal projections. Define

Then, for every vector \(v\),

Let \(M\) be a commutative monoid. If \(L\leq N\), \(i\in \{ 0,\ldots ,N{-}1\} \), and \(f\colon \{ 0,\ldots ,N{-}1\} \to M\), then the product of \(f\) over the chain equals the product over the cyclic window beginning at \(i\) times the product over its ordered complement.

Suppose every bond dimension is positive. The transfer map of \(\bigoplus _k B_k\) is idempotent if and only if, for every pair \(j,j'\), the mixed transfer operator is idempotent:

Let \(1\leq R\leq m\), \(2m\leq N\), and let \(h_i\) be cyclically indexed orthogonal projections such that \(h_i\) commutes with \(h_{i+d}\) whenever \(R\leq d\leq N-R\). Suppose that, for every \(s\) and \(v\),

If \((R-1)^2{\lt}m\gamma \), then, with

one has

The coefficient for general \(R\) is derived here and is not attributed to the cited papers.

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.

Let \(A\) be primitive with positive-definite fixed point \(\rho \) and nonzero bond dimension. There are constants \(C,c,r{\gt}0\), with \(r{\lt}1\), such that for every spectator length \(K\) and every \(l\geq 1\) satisfying \(cr^l{\lt}1\),

The constants are independent of \(K\).

Let \(U\) and \(V\) be ground-space ranges in a finite-dimensional complex Hilbert space, let \(W=U\cap V\), and let \(q_U\) and \(q_V\) be the orthogonal projectors onto \(U\) and \(V\). Suppose that \(\eta \ge 0\) and

whenever \(x\in U\cap W^\perp \) and \(y\in V\cap W^\perp \). Define the complementary excitation projections by \(h_U=\mathbb {1}-q_U\) and \(h_V=\mathbb {1}-q_V\). Then, for every vector \(v\),

If \(A\) and \(B\) are gauge equivalent and \(A\) is normal, then \(B\) is normal.

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 \(n\in \mathbb {N}\).

Let \(c{\gt}0\), let \(0{\lt}r{\lt}1\), and let \(l,K\in \mathbb N\). If \(cr^l{\lt}1\), then

If the ground-space map \(\Gamma _n^{\mathrm{ES}}\) is injective, then its Gram endomorphism \(\mathcal K_n=(\Gamma _n^{\mathrm{ES}})^\dagger \Gamma _n^{\mathrm{ES}}\) is invertible.

If \(A\) is injective and \(L \ge 2\), then the nontrivial inclusion in the intersection property holds:

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 \(X\in M_{D}(\mathbb {C})\), \(\Gamma _L^{\mathrm{ES}}(U(X))=\iota _L(\Gamma _L(X))\).

If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then

where \(Q_{AX}=\mathbb {1}-P_{AX}\) and \(Q_{XB}=\mathbb {1}-P_{XB}\) are the complementary projectors. This is the algebraic identity obtained from the complementary projectors in [ CPGSV16 , Appendix D, Section D.2 ] ; it is not the full source parent commuting Hamiltonian condition.

Corollary of Theorem 13.2.33: if \(P_K\) has idempotent-product data, the \(A\)-observables commute with \(P_{XB}\), and the \(B\)-observables commute with \(P_{AX}\), then regions \(A\) and \(B\) are decorrelated w.r.t. \(K\).

If \(P_K\circ P_K=P_K\), then \(P_K\) has idempotent-product data.

If \(T\) is an injective map from a finite-dimensional Hilbert space into a complete Hilbert space, then

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

Assume \(D\geq 1\) and let \(\rho \in M_{D}(\mathbb {C})\) be positive definite. Set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). If \(\Gamma _n^{\mathrm{ES}}\) is injective and \(S_n=\mathcal K_n^{-1}\), then

For every prefix \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) and suffix \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \),

If the length-\(K\) ground-space map is injective, then the left boundary map with any spectator length \(L\) is injective.

The standard length-\((K+L)\) boundary map factors through the left boundary map:

for every \(X\in M_{D}(\mathbb {C})\).

The range of the left boundary map is exactly the left ground space:

At \(L=1\), transport of the left boundary map range by the inverse canonical \(\ell ^2\) isomorphism identifies it with the open-chain left ground submodule:

Concretely, a vector \(v\) in the \((K+1)\)-site Euclidean space belongs to \(\mathcal U_K(A)\) exactly when the corresponding function satisfies the left ground condition (every fixed-final-site restriction lies in \(G_K(A)\); Definition 12.4.1.4).

A normal left-canonical renormalization fixed-point tensor of nonzero virtual dimension has pairwise commuting length-two parent interactions on every periodic chain of length \(N{\gt}2\), and its periodic matrix product vector is annihilated by every such interaction. The removal of the left-canonical hypothesis and the ground-space spanning equation in [ CPGSV16 , Definition 3.9 ] remain separate.

Suppose that \(\Gamma _K^{\mathrm{ES}}\) is injective and write \(S_K=\mathcal K_K^{-1}\). Then

If \(\Gamma _K^{\mathrm{ES}}\) is injective, then

For arbitrary virtual dimension, the adjoints of the tail and left virtual maps are

If, in addition, \(D{\gt}0\) and \(\rho \in M_{D}(\mathbb {C})\) is positive definite, set \(\mathcal K_\infty =\mathscr R(P_\rho )\) and \(S_\infty =\mathcal K_\infty ^{-1}\). Then

Put \(q(A)=1-P_{G_2(A)}\) for the canonical two-site parent interaction. On the three-site space \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\), with sites \(0,1,2\) labelled \(A,X,B\), the two adjacent translated length-two parent interactions are the two local actions

This identifies the two translated parent interactions with the two supports appearing in [ CPGSV16 , Definition D.2 ] ; it does not assert that they commute.

Suppose the canonical two-site parent interaction \(q(A)=1-P_{G_2(A)}\) satisfies the overlapping \(AX/XB\) commutation condition of Definition 13.1.2. Then, on the three-site chain, \(h^{(3)}_0(A,2)h^{(3)}_1(A,2) = h^{(3)}_1(A,2)h^{(3)}_0(A,2)\). This only identifies the first two translated local terms with the lifted \(AX\) and \(XB\) actions; the construction of the [ CPGSV16 , Appendix D.2 ] projectors associated with the [ CPGSV16 , Appendix B ] basic-vector form is a separate step.

Suppose the Appendix D.2 projectors \(Q_{AX}\) and \(Q_{XB}\) satisfy the local parent-commuting condition on \(A X B\), and suppose that the canonical two-site parent interaction is their complement on both faces: \(q(A)=1-Q_{AX}=1-Q_{XB}\). Then the first two translated length-two local terms on the three-site chain commute, \(h^{(3)}_0(A,2)h^{(3)}_1(A,2) = h^{(3)}_1(A,2)h^{(3)}_0(A,2)\).

Suppose the projectors \(Q_{AX}\) and \(Q_{XB}\) satisfy the [ CPGSV16 , Definition D.2 ] local parent-commuting condition on \(A X B\), and suppose that the canonical two-site parent interaction is identified, as a two-site operator, with the two source projectors: \(q(A)=Q_{AX}\) and \(q(A)=Q_{XB}\). After lifting these two equalities to the \(AX\) and \(XB\) faces, the first two translated length-two local terms on the three-site chain commute. Thus, \(h^{(3)}_0(A,2)h^{(3)}_1(A,2) = h^{(3)}_1(A,2)h^{(3)}_0(A,2)\).

Let \(Q_{AX}\) and \(Q_{XB}\) be two-site projectors whose lifted \(AX\) and \(XB\) actions commute. Suppose that \(q(A)=Q_{AX}\) and \(q(A)=Q_{XB}\). Then the first two translated length-two local terms on the three-site chain commute.

Let \(P_i\) be a finite family of symmetric projections on a finite-dimensional complex Hilbert space, let \(H=\sum _iP_i\), and let \(0{\lt}\gamma \le 1\). Suppose that a real coefficient matrix \((c_{ij})\) has row and column sums at most one away from the diagonal, and that, for \(i\ne j\) and every \(v\),

Then

for every \(v\). Consequently, for every \(v\in (\ker H)^\perp \),

Let \(A\) be a primitive MPS tensor with nonzero physical and bond dimensions, and suppose that its fixed point is positive definite. There is a blocking length \(p{\gt}0\) such that \(A\) is \(p\)-block injective and, with \(B=\operatorname {block}_p(A)\), for every \(N\geq 4\) and every \(v\in (\ker H_{N,2}(B))^\perp \),

Under the hypotheses of Theorem 13.3.5.63, one can choose the same \(p{\gt}0\) so that the original tensor has interaction range \(2p\) and, for every \(N\ge 4\) and every \(v\in (\ker H_{Np,2p}(A))^\perp \),

If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then

This is the algebraic consequence corresponding to equation (D.2) from [ CPGSV16 ] : \(K_{AXB}=(K_{AX}\otimes H_B)\cap (H_A\otimes K_{XB})\). The source equation itself also carries the tensor-product local-subspace hypotheses.

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 \(G_L(A)\).

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 \(G_L(A)\).

Let \(A\) be a primitive MPS tensor with nonzero bond dimension and a positive-definite fixed point. There are an overlap length \(l{\gt}1\), measured in sites of \(A\), and a number \(\varepsilon _l\geq 0\) such that \(A\) is \(l\)-block injective, the projector defect is bounded uniformly in the left length \(K\) by \(\varepsilon _l\), and

Nachtergaele’s projector-defect identity following condition C3 [ Nac96 , Eq. (2.4) ] , together with the two-projection estimate, gives the corresponding anticommutator bound for every \(K\). If \(A\) is already a blocked representative, one site of \(A\) is one blocked-chain filtration step.

Assume \(D\ge 1\) and let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Set \(n=K+L_0\) and \(l=L_0\), and write

Thus \(S\) and \(T\) are the left and tail ground spaces in the Hilbert space on \(\Lambda _{n+1}\). Let \(q_S=P_{S^\perp }\) and \(q_T=P_{T^\perp }\) be the complementary excitation projections. If

then every vector \(v\) in the Hilbert space on \(\Lambda _{n+1}\) satisfies

Under the hypotheses of Theorem 13.3.5.67 with \(R=2\), if \(1{\lt}m\gamma \), then

Equivalently, the open-window threshold is \(\gamma {\gt}1/m\). This is Knabe’s nearest-neighbor threshold  [ Kna88 ] , quoted in [ PGVWC07 , lines 1483–1489 ] and [ CPGSV21 , lines 2194–2197 ] .

Suppose that the canonical length-two parent terms of \(A\) commute on a three-site chain. Then the complementary two-site interaction admits a finite sector graph as in Definition 13.1.75.

Let \(P\) be a sector decomposition with representatives \(A_1,\ldots ,A_g\) and total tensor \(B\). Suppose that the vectors \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\) are linearly independent for all sufficiently large \(N\), and that \(B\) satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition with these representatives. Then there is \(N_0\) such that, for every \(N{\gt}N_0\), \(\dim _{\mathbb {C}}\ker H_2^{(N)}(B) = g\). Thus the ground-state degeneracy is eventually constant, as required in [ Bei12 , Section IV ] .

Let \(B\) be normal and in left-canonical form. Suppose that the family \(A_j\) is a BNT for \(B\), and that the nearest-neighbor parent Hamiltonian of \(B\) has the all-chain ground-space property of Definition 13.1.18: for every \(N{\gt}2\), the length-two translated parent interactions commute, \(h_i(B,2)h_j(B,2)=h_j(B,2)h_i(B,2)\), the vector \(V^{(N)}(B)\) has zero energy, and \(\ker H_2^{(N)}(B) = \operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \). Then \(B\) is a renormalization fixed point.

This is the normal left-canonical representative case of the reverse implication in Theorem 3.10 of [ CPGSV16 ] . The source theorem is stated for a tensor in canonical form; the unrestricted canonical-form statement is not asserted here.

Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and set

Suppose that, for every \(N{\gt}2\), the translated length-two parent interactions of \(B\) commute, every \(V^{(N)}(A_j)\) has zero energy, and

Then \(B\) is a renormalization fixed point.

This is the multiplicity-one, unit-weight representative case of the reverse implication in Theorem 3.10 of [ CPGSV16 ] . It does not assert the printed theorem for repeated copies or arbitrary raw weights.

If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then \(A\) is normal (has eventually full Kraus rank). No identity-in-the-one-step-span condition is needed.

This connects the complementary-gap condition in Definition 7.6.1 directly to normality.

Let \(A\) be a normal tensor. There exist an interaction range \(L{\gt}1\) and a constant \(\gamma {\gt}0\) such that, for every \(N\ge 2L\) and every \(v\in (\ker H_{N,L}(A))^\perp \),

Theorem 13.3.5.74 proves the estimate with \(L=2p\) when the periodic chain length is divisible by \(p\). The unrestricted statement still requires a remainder-length comparison for the finitely many congruence classes modulo \(p\).

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 \(G_N(A)\).

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 \(\Lambda _{n+1}\).

For every \(A\), \(L\), and \(N\),

Assume \(D\geq 1\), \(L_0{\gt}0\), and \(N\geq L_0+1\). If \(A\) is \(L_0\)-block-injective, then

If \(L{\gt}0\), then

If \(L{\gt}0\) and \(v\in (\ker H^{\mathrm{open}}_{L,L}(A))^\perp \), then

In particular,

Thus \(\gamma _{l+1}=1\) at the full-window volume \(N=L=l+1\) in [ Nac96 , Theorem 2.1(i) ] . This neither asserts that the excited subspace is nonzero nor supplies a gap for any \(N{\gt}L\).

Assume \(D\geq 1\), \(L_0{\gt}0\), and \(N\geq L_0+1\). If \(A\) is \(L_0\)-block-injective, then

This is the canonical MPS instance of the ground-state property in [ Nac96 , Equation (3.13) ] . The right-hand side is the open boundary-condition space and may therefore be degenerate.

If \(H^{\mathrm{open}}_{N,L}(A)v=0\), then, for every \(i\in I_{L,N}\),

Let \(A\) be a tensor and let \(L{\gt}1\). Suppose that, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,

Then the parent Hamiltonian has the explicit gap constant \(\gamma _L:=1/(4L){\gt}0\). Concretely, if \(N\ge 2L\) and \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), then \(\gamma _L\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).

Let \(L{\gt}1\). Put \(m=2(L-1)\). Suppose that there is a constant \(\eta \ge 0\) such that \(\eta m{\lt}1\), and suppose that, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,

Then the parent Hamiltonian has the gap constant \(1-\eta m{\gt}0\). Concretely, if \(N\ge 2L\) and \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), then \((1-\eta m)\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).

Let \(L{\gt}1\). Put \(m=2(L-1)\). Suppose there is a constant \(\eta \ge 0\) with \(\eta m{\lt}1\) and that, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,

that is, a bound \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| \le \eta \) on the operator-product norm rather than the directional bound of Theorem 13.3.5.78. Then the parent Hamiltonian has the gap constant \(1-\eta m{\gt}0\). Concretely, if \(N\ge 2L\) and \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), then \((1-\eta m)\| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). This is a conditional reduction. The local terms \(h_{i,\mathrm{ES}}=p_i\) are the excitation projections. If their overlapping ranges have a common nonzero vector, then \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| =1\), so the hypothesis is unsatisfiable for \(\eta {\lt}1\) in that case. In degenerate cases, such as a zero excitation projection, the operator-product norm can be smaller. In any case this is not the quantity controlled by the cited principal-angle estimates. The comparison is documented in [ con26m ] .

Let \(\mathcal{H}=\mathbb {C}^d\), and let \(Q\) be an endomorphism of \(\mathcal{H}\otimes \mathcal{H}\). Under the canonical identifications of two- and three-site configurations with pairs and right-associated triples, the matrices of its two adjacent actions are

Equivalently, for all \(a,x,b,a',x',b'\in \{ 0,\ldots ,d{-}1\} \),

This is the homogeneous three-single-site specialization of the factor actions on \(\mathcal{H}_A\otimes \mathcal{H}_X\otimes \mathcal{H}_B\) in [ CPGSV16 , Definition D.2 ] ; here \(\mathcal{H}_A=\mathcal{H}_X=\mathcal{H}_B=\mathcal{H}\). No assertion is made for unequal factor spaces.

Under the canonical \(\ell ^2\) identification of the \(N\)-site state space, the parent ground space is exactly the kernel of the corresponding parent Hamiltonian.

For a tensor \(A\) and a range \(L{\gt}1\), assume either the cyclic-window anticommutator estimate of Theorem 13.3.5.77 or the stronger overlapping cyclic-window norm-compression estimate of Theorem 13.3.5.76. Then there exists \(\gamma {\gt}0\) such that for every \(N\ge 2L\) and every vector \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \) one has \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). By Lemma 13.3.20, this implies the usual uniform lower bound on every nonzero eigenvalue of the parent Hamiltonian.

For a tensor \(A\) and a range \(L{\gt}1\), suppose there is a constant \(\eta \ge 0\) such that \(\eta \, 2(L-1){\lt}1\) and, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,

Then there exists \(\gamma {\gt}0\) such that, for every \(N\ge 2L\) and every vector \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \).

For a tensor \(A\) and a range \(L{\gt}1\), suppose there is a constant \(\eta \ge 0\) such that \(\eta \, 2(L-1){\lt}1\) and, for every \(N\ge 2L\) and every overlapping off-diagonal pair of cyclic length-\(L\) windows,

Then there exists \(\gamma {\gt}0\) such that, for every \(N\ge 2L\) and every vector \(v\in (\mathcal G_{N,L}^{\mathrm{ES}}(A))^\perp \), \(\gamma \| v\| \le \| H_{N,L}^{\mathrm{ES}}(A)v\| \). As in Theorem 13.3.5.79, this is a conditional reduction: the operator-product hypothesis is unsatisfiable for any overlapping pair whose excitation ranges have a common nonzero vector, since then \(\| h_{i,\mathrm{ES}}h_{j,\mathrm{ES}}\| =1\). The statement does not assert this norm identity in degenerate cases.

Assume \(N{\gt}0\) and \(L\le N\). Then \(\ker H_N(A,L)=G_L^{(N)}(A)\). Equivalently, after the canonical Hilbert-space identification, \(\mathcal K_N^{\mathrm{ES}}(A,L)=e_N^{-1}G_L^{(N)}(A)\), where \(\mathcal K_N^{\mathrm{ES}}(A,L)\) is the zero-energy space of \(H_{N,L}^{\mathrm{ES}}(A)\).

Assume \(N{\gt}0\) and \(L\le N\). If a vector \(\psi \) has zero energy for the finite periodic parent Hamiltonian, \(H_N(A,L)\psi =0\), then every cyclic length-\(L\) restriction of \(\psi \) lies in the local MPS ground space. Equivalently, \(\ker H_N(A,L)\subseteq G_L^{(N)}(A)\).

If \(\rho =\rho ^\dagger \), then \(\operatorname{tr}_L\rho =(\operatorname{tr}_L\rho )^\dagger \).

If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_{AX}\circ P_K=P_K\).

The transfer map of an MPS tensor is idempotent if and only if the tensor has a physical blocking isometry.

If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_K\circ P_{AX}=P_K\).

If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_K\circ P_{XB}=P_K\).

If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_K\circ P_K=P_K\).

Restricting the left boundary map to the prefix for a fixed suffix \(\tau \in \{ 0,\ldots ,d{-}1\} ^L\) recovers \(\Gamma _K(Z_\tau )\):

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 \(n\geq 1\),

Under the hypotheses of Theorem 12.3.35, there are constants \(g,c,r{\gt}0\) with \(r{\lt}1\) and \(c=\lVert \mathcal I_\infty \rVert g\) such that (42) 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

Let \(A\) be a primitive MPS tensor with nonzero virtual dimension, and let \(J_K=J^{\mathrm{tail}}_K\) be its length-\(K\) tail virtual map. Then there is a constant \(C{\gt}0\) such that, for every \(K\in \mathbb {N}\),

No positive-definiteness hypothesis on the fixed point is required.

Let \(A\) be a primitive MPS tensor with positive-definite fixed point \(\rho \). Then \(A\) satisfies the left-canonical centered residual estimate of Theorem 13.3.5.29.

For the convention that an \(L\)-site interaction is zero when \(L{\gt}N\), the two-site parent Hamiltonian on a one-site periodic chain satisfies

This convention is local to the present development. The two-site terms in [ Bei12 , Section III ] are not assigned an action when both site labels denote the same site. The nearest-neighbor condition in [ CPGSV16 , Theorem 3.10(iii) ] , source lines 534–540, only concerns \(N{\gt}2\).

Let \(P\) and \(Q\) be orthogonal projections on a real or complex Hilbert space. For every vector \(v\),

Let \(P_U\), \(P_V\), and \(P_{U\cap V}\) be the orthogonal projections onto \(U\), \(V\), and their intersection. If

then the Friedrichs overlap of \(U\) and \(V\) is at most \(\eta \).

If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_{XB}\circ P_K=P_K\).

The range of \(\Gamma _L^{\mathrm{ES}}\) is exactly \(G_L^{\mathrm{ES}}(A)\).

The reassociated tail map satisfies

Reassociation preserves the tail Gram:

If \(\Gamma _{L+Q}^{\mathrm{ES}}\) is injective, then the inverse Gram of the reassociated tail map is the fiberwise direct sum of \((\mathcal K_{L+Q})^{-1}\).

If \(\Gamma _{L+Q}^{\mathrm{ES}}\) is injective, then

The common physical map is an isometry, \(V^*V=1\). Moreover, the equal-sector bond embedding and the physical support projector satisfy

and \(P_{2,\mathrm{res}} = 1-q_2\! \left(\bigoplus _j B_j\right)\). If the \(B_j\) form a basis of normal tensors and their direct sum is a renormalization fixed point, the sectorwise Appendix B decompositions and the common residual isometry required above exist.

Let \(\mathcal V=\bigsqcup _j\{ j\} \times [D_j]\times [D_j]\) be the disjoint union of the within-sector virtual pairs. For \(x=(j,\alpha ,\beta )\) in \(\mathcal V\), set \(u_x(i)=(U_j^i)_{\alpha ,\beta }\). Then \(\sum _i u_x(i)\overline{u_y(i)}=\delta _{x,y}\).

If \(P_K=P_{AX}\circ P_{XB}\) is idempotent-product data, then \(P_{XB}\circ P_{AX}=P_K\).

If \(A\) has nonzero virtual dimension, is a normal tensor in the sense of [ CPGSV16 ] , and is a renormalization fixed point, then for every chain length \(N{\gt}2\) the two-site local terms commute. This theorem records the commutation part of the nearest-neighbor source condition, not the ground-space spanning part.

If \(A\) has nonzero virtual dimension, is a normal tensor in the sense of [ CPGSV16 ] , and is a renormalization fixed point, then for every chain length \(N{\gt}2\) the MPS vector \(V^{(N)}(A)\) satisfies the commutation and zero-energy equations for nearest-neighbor local terms:

The zero-energy equation is the usual parent-Hamiltonian frustration-freeness statement. The separate ground-space spanning clause of the source definition is not asserted here.

Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\) its translated two-site parent interactions commute and

Hence \(B\) satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition. This is the multiplicity-one distinct-sector forward implication in [ CPGSV16 , Definition 3.9 and Theorem 3.10 ] . It does not include repeated copies or arbitrary raw sector weights.

Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and set \(A=\bigoplus _{j=1}^g A_j\). Then

This statement concerns the multiplicity-one, unit-weight direct sum; it does not include repeated copies or arbitrary raw weights.

Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\) its translated two-site parent interactions commute pairwise. Thus \(B\) satisfies the nearest-neighbor commuting parent-Hamiltonian condition.

This statement treats one copy of every distinct sector. It does not include repeated copies or arbitrary raw sector weights.

Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\),

This is the multiplicity-one distinct-sector case of [ CPGSV16 , Theorem 3.10 ] . It does not include repeated copies of a basis tensor or arbitrary raw sector weights.

If \(A\) is a normal RFP tensor, then \(A\) is injective.

Let \(A\) have positive bond dimension and be a normal renormalization fixed-point tensor in the sense of [ CPGSV16 ] . If the canonical-form tensor consists of the single normal sector \(A\), then for every \(N{\gt}2\), \(\ker H_2^{(N)}(A) =\operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \). Its translated two-site interactions also commute and annihilate \(V^{(N)}(A)\). Hence the singleton family satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition.

This statement is restricted to one normal sector. It does not assert the corresponding result for canonical-form tensors with several sectors or repeated copies.

The tail Gram is the direct sum over prefix configurations of copies of \(\mathcal K_L\), and the left Gram is the direct sum over suffix configurations of copies of \(\mathcal K_K\). Whenever the corresponding local boundary map is injective, the inverse Gram is the direct sum of the local inverse Grams. These identities include empty spectator index types and assert no norm estimate.

Equip each virtual matrix fiber with the unweighted Hilbert–Schmidt inner product and flatten the finite family of fibers into a Euclidean space. The resulting continuous tail and left boundary maps have ranges \(\mathcal V_{K,L}(A)\) and, at suffix length one, \(\mathcal U_K(A)\). Moreover, if \(J^{\mathrm{tail}}_K(X)_u=XA^u\) and \(J^{\mathrm{left}}_L(X)_\tau =A^\tau X\), then

Let \(q\in \mathbb N\), let \(X=(X_u)_{u\in \{ 0,\ldots ,d{-}1\} ^K}\), and let \(Z=(Z_j)_{j\in \{ 0,\ldots ,d{-}1\} ^q}\) be virtual boundary-matrix families. After the canonical identification of \(K+(l+q)\) with \((K+l)+q\), the mixed Gram for the tail window of length \(l+q\) and the left window with a \(q\)-site increment is

where \(U\) is Frobenius vectorization and \(\mathcal K_l=\Gamma _l^\dagger \Gamma _l\). The multiplication order is fixed by \(\operatorname{tr}(A^{\tau ,j}X_u)=\operatorname{tr}(A^\tau (A^jX_u))\) and \(\operatorname{tr}(A^{u,\tau }Z_j)=\operatorname{tr}(A^\tau (Z_jA^u))\). The one-site identity is the specialization \(q=1\). This is the exact algebraic whole-increment identity relevant to condition C3\('\) in [ Nac96 , Equation (2.5) ] ; it does not prove the norm estimate, uniformity in \(q\), or any numerical contraction required there.

Specialize the preceding theorem to \(q=1\). Let \(X=(X_u)_{u\in \{ 0,\ldots ,d{-}1\} ^K}\) and \(Z=(Z_j)_{j\in \{ 0,\ldots ,d{-}1\} ^1}\) be the resulting boundary families. If \(\rho \in M_{D}(\mathbb {C})\) has nonzero trace and \(\mathcal K_\infty =\mathscr R(P_\rho )\), then

Here \(\mathcal K_\infty U(Z_jA^u) =U((\operatorname{tr}\rho )^{-1}Z_jA^u\rho )\), so the limiting multiplication is on the right and is generally not the identity.

Let \(I\) be a finite index set and let \(A=(A_{ij})_{i,j\in I}\) be a complex matrix. Then

For every prefix \(u\colon \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \) and suffix \(\tau \colon \{ 0,\ldots ,L{-}1\} \to \{ 0,\ldots ,d{-}1\} \),

The standard length-\((K+L)\) boundary map factors through the tail boundary map:

for every \(X\in M_{D}(\mathbb {C})\).

The range of the tail boundary map is exactly the tail ground space:

For every \(K,L\in \mathbb {N}\), transport of the tail boundary map range by the inverse canonical \(\ell ^2\) isomorphism identifies it with the open-chain tail ground submodule:

Concretely, a vector \(v\) in the \((K+L)\)-site Euclidean space belongs to \(\mathcal V_{K,L}(A)\) exactly when the corresponding function satisfies the tail ground condition (Definition 12.4.5.6).

Restricting the tail boundary map to the suffix for a fixed prefix \(u\in \{ 0,\ldots ,d{-}1\} ^K\) recovers \(\Gamma _L(Y_u)\):

If \(\Gamma _L^{\mathrm{ES}}\) is injective, then

Under the same hypotheses, there is an integer \(p{\gt}0\), measured in sites of \(A\), such that \(A\) is \(p\)-block injective and

Fix a tensor \(A\) with physical dimension \(d{\gt}0\) and a coefficient \(c\in \mathbb R\). For every chain length \(N\) and site \(i\in \{ 0,\ldots ,N{-}1\} \), set \(h_i^{(N)}:=h_{i,\mathrm{ES}}^{(N,2)}(A)\), the Euclidean range-two parent term of this same tensor \(A\) at \(i\). Suppose that, for every three-site vector \(w\),

Then, for every \(N\geq 3\), every site \(i\in \{ 0,\ldots ,N{-}1\} \), and every \(N\)-site vector \(v\), the forward pair satisfies

and the backward pair satisfies

All site indices are taken modulo \(N\), so these conclusions include pairs and three-site windows crossing the periodic cut.

Suppose the first two length-two parent interactions commute on the three-site chain: \(h_0^{(3)}h_1^{(3)} =h_1^{(3)}h_0^{(3)}\). Then, for every \(N{\gt}2\) and every \(i\in \mathbb Z/N\mathbb Z\), the two adjacent translates commute: \(h_i^{(N)}h_{i+1}^{(N)} =h_{i+1}^{(N)}h_i^{(N)}\). For each exterior configuration \(\tau \), let \(R_{i,\tau }^{(3)}\) restrict to the cyclic three-site window beginning at \(i\). It intertwines the two \(N\)-site terms with the two three-site terms:

Thus the same local \(A\! X\! B\) calculation is placed at every cyclic translate through the two restriction identities above. The hypothesis \(N{\gt}2\) is essential to this transport: at \(N=2\) the two cyclic windows contain the same sites in opposite orders. This case is not part of [ CPGSV16 , Theorem 3.10 ] .

If \(A\) is injective, then \(G_3(A) = \ker (q_2(A)_{AX}) \cap \ker (q_2(A)_{XB})\). More generally, on \(L+1\) sites, simultaneous annihilation by the two adjacent length-\(L\) parent interactions is equivalent to membership in \(G_{L+1}(A)\) whenever \(A\) is injective and \(L{\gt}1\).

For a nonempty closed chain, the trace of the word \(A^{\sigma _0}\cdots A^{\sigma _{L-1}}\) is the sum over cyclic virtual-index configurations:

For \(P\geq 0\), \(\widehat P_{AX}P=P=P\widehat P_{AX}\).

If \(P=P^\dagger \) and \(A\) and \(B\) are decorrelated, then

If \(P=P^\dagger \) and \(A\) and \(B\) are decorrelated, then, for arbitrary complex matrices \(O_A\) and \(O_B\), one has \(P O_B(\mathbb {1}-P)O_A P=0\).

Let \(P\), \(P_{AX}\), and \(P_{XB}\) be orthogonal projectors, and write their lifts to the tripartite space as \(\widehat P_{AX}=P_{AX}\otimes \mathbb {1}_B\) and \(\widehat P_{XB}=\mathbb {1}_A\otimes P_{XB}\). Suppose that the lifted projectors commute, \(\widehat P_{AX}\widehat P_{XB} =\widehat P_{XB}\widehat P_{AX}\). Then

This is the passage from [ CPGSV16 , Definition D.2, lines 2205–2218 ] to the projector identity at lines 2279–2289.

If \(A\) and \(B\) are decorrelated, then \(\widehat\rho _{XB}(\mathbb {1}-P)\widehat\rho _{AX}=0\).

For matrix units on \(A\) and \(B\),

Observable decorrelation holds if and only if its complex-linear extension holds.

Let \(P=P^\dagger =P^2\). Then \(P\) is the ground-space projector of a parent commuting Hamiltonian on \(AX\) and \(XB\) if and only if regions \(A\) and \(B\) satisfy complex-linear decorrelation.

Let \(P=P^\dagger =P^2\). Then \(P\) is the ground-space projector of a parent commuting Hamiltonian on \(AX\) and \(XB\) if and only if regions \(A\) and \(B\) are decorrelated when tested on Hermitian observables. This is [ CPGSV16 , Proposition D.3, lines 2221–2289 ] .

If \(P=P^\dagger \) and \(A\) and \(B\) are decorrelated, then \(\widehat\rho _{AX}(\mathbb {1}-P)\widehat\rho _{XB}=0\).

If \(P=P^\dagger =P^2\) and \(A\) and \(B\) are decorrelated relative to \(P\), then

For \(P\geq 0\), \(\widehat P_{XB}P=P=P\widehat P_{XB}\).

Suppose that the canonical length-two parent terms of \(A\) commute on a three-site chain. Then \(Q_A\) is Hermitian, and its natural placements on the first and last pairs of a three-fold tensor product commute:

Under the preceding hypothesis, there are positive integers \(d_q,m_q\), a unitary identification

and Hermitian matrices \(R_q\) on \(\mathcal H\otimes \mathcal H_{q,l}\) and \(S_q\) on \(\mathcal H_{q,r}\otimes \mathcal H\) such that, after conjugating the middle site by this unitary, the first identity below holds on the first pair and the second on the last pair:

This is the spatial decomposition of [ Bei12 , Lemma 2.1 ] for the canonical parent interaction.

Let \(A\) be primitive with positive-definite fixed point \(\rho \) and nonzero bond dimension. There is an integer \(p{\gt}0\), measured in sites of \(A\), such that \(A\) is \(p\)-block injective and, for every \(K\in \mathbb N\),

Let \((P_i)_{i\in I}\) be a finite family of positive operators on a finite-dimensional complex inner product space, with real weights \(0{\lt}w_i\leq 1\), and let \(\gamma \geq 0\). If

for every \(v\in \bigl(\ker \sum _{i\in I}w_iP_i\bigr)^\perp \), then

for every \(v\in \bigl(\ker \sum _{i\in I}P_i\bigr)^\perp \).

Let \((P_i)_{i\in I}\) be a finite family of positive operators on an inner product space. A positive operator \(P\) annihilates a vector \(v\) as soon as \(\operatorname {Re}\langle Pv,v\rangle =0\), and consequently, for real weights \(w_i{\gt}0\),

equivalently \((\sum _{i\in I}w_iP_i)v=0\) exactly when \(P_iv=0\) for every \(i\in I\). For the empty family both sums are zero and all three kernels are the full space. This is the analytic kernel step of the blocked-sum comparison in Lemma block-obc of Rozmán–Molnár–Schuch ¹ , in which the blocked parent Hamiltonian is a weighted sum \(\sum _i\gamma _ih_i\) of positive local terms with \(\gamma _i{\gt}0\).

Let \((P_i)_{i\in I}\) be a finite family of positive operators on an inner product space, with real weights \((w_i)_{i\in I}\). If \(w_i\geq 0\) for every \(i\), then \(\sum _{i\in I}w_iP_i\) is positive. If \(w_i\leq 1\) for every \(i\), then

in Loewner order. This is the comparison step of the blocked-sum argument in the source cited in Theorem 13.3.5.70, where the coefficients satisfy \(0{\lt}\gamma _i\leq 1\).

The centered residual factors as

Assume \(D{\gt}0\), let \(\rho \) be positive definite, and suppose that \(\Gamma _{L+Q}^{\mathrm{ES}}\) and \(\Gamma _{K+L}^{\mathrm{ES}}\) are injective. Put

Then the centered physical sandwich \(\widehat R_{K,L,Q}\) satisfies

The factorization gives

If \(A\) is left-canonical, then

Assume \(D{\gt}0\) and let \(\rho \) be positive definite. Put \(I_t=\mathcal K_{L+Q}^{-1}\), \(I_f=\mathcal K_{K+L+Q}^{-1}\), \(I_\ell =\mathcal K_{K+L}^{-1}\), and \(I_\infty =\mathcal K_\infty ^{-1}\). Under the injectivity assumptions required to define the indicated inverse Grams, the tail, full, and left corrections \(C_t,C_f,C_\ell \) satisfy

The full correction satisfies

The left correction satisfies

If \(\Gamma _{K+L}^{\mathrm{ES}}\) and \(\Gamma _{L+Q}^{\mathrm{ES}}\) are injective, respectively, then

Moreover,

Assume \(D{\gt}0\), let \(\rho \) be positive definite, and suppose that \(\Gamma _{L+Q}^{\mathrm{ES}}\), \(\Gamma _{K+L}^{\mathrm{ES}}\), and \(\Gamma _{K+L+Q}^{\mathrm{ES}}\) are injective. Then the whole-increment projector defect is the centered physical sandwich minus the tail Gram correction, the full inverse-Gram correction, and the left Gram correction.

Let \(A\) be primitive with positive-definite fixed point \(\rho \) and nonzero bond dimension. There are constants \(C,c,r{\gt}0\), with \(r{\lt}1\), such that for all \(K,L,Q\in \mathbb N\) with \(L\geq 1\) and \(cr^L{\lt}1\),

Here \(P^{\mathrm{tail}}_{K;L,Q}\) is the range projector of the reassociated tail boundary map and \(P^{\mathrm{left}}_{K+L,Q}\) is the range projector of the left boundary map. The constants are independent of both the prefix length \(K\) and the suffix length \(Q\).

For every \(K,L,Q\in \mathbb {N}\),

The matrix fiber of \(F_{K,Q}X\) at \((u,j)\) is \(A^jX_u\).

If \(A\) is left-canonical, then the length-\(Q\) left virtual map \(J_Q^{\mathrm{left}}\) preserves the Hilbert–Schmidt norm. In particular,

Assume \(D{\gt}0\). For a positive-definite matrix \(\rho \), the product \(F_{K,Q}^\dagger \mathcal K_\infty ^{\oplus }G_{K,Q}\) equals the corresponding virtual-word expression formed from \(J_K^{\mathrm{tail}}\), \((J_Q^{\mathrm{left}})^\dagger \), \(\mathcal K_\infty \), and \(\mathcal K_\infty ^{-1}\).

With the notation of Theorem 13.3.3.22,

Let \(\rho \in M_{D}(\mathbb {C})\) have nonzero trace and set \(\mathcal K_\infty =\mathscr R(P_\rho )\). Then

The overlap factors satisfy

Let \(T=\widetilde\Gamma ^{\mathrm{tail}}_{K;L,Q}\), \(S=\Gamma ^{\mathrm{left}}_{K+L,Q}\), and \(C=\Gamma _{K+L+Q}^{\mathrm{ES}}\). Assume separately that \(T\), \(S\), and \(C\) are injective. Since \(TJ_K^{\mathrm{tail}}=C=SJ_Q^{\mathrm{left}}\), the difference \(P_TP_S-P_C\) equals the exact inverse-Gram residual obtained from these two factorizations.

The matrix fiber of \(G_{K,Q}Z\) at \((u,j)\) is \(Z_jA^u\).

Assume \(D{\gt}0\), let \(\rho \) be positive definite, and assume that \(\Gamma _{K+L+Q}^{\mathrm{ES}}\) is injective. Write \(K_t=\mathcal K_{L+Q}^{\oplus \Omega _K}\), \(K_t^\infty =\mathcal K_\infty ^{\oplus \Omega _K}\), \(K_\ell =\mathcal K_{K+L}^{\oplus \Omega _Q}\), \(K_\ell ^\infty =\mathcal K_\infty ^{\oplus \Omega _Q}\), \(I_f=\mathcal K_{K+L+Q}^{-1}\), \(I_\infty =\mathcal K_\infty ^{-1}\), \(J_t=J_K^{\mathrm{tail}}\), and \(J_\ell ^\dagger =(J_Q^{\mathrm{left}})^\dagger \). Then the exact virtual residual satisfies

If \(A\) is left-canonical, then for every \(Q\in \mathbb {N}\),

where \(\Omega _Q=\{ 0,\ldots ,({-}1\} d^Q)\) is the set of physical words of length \(Q\).

For every \(n\),

Consequently, if \(L{\gt}0\) and \(A\) is \(L\)-block injective, then \(A\) is \(m\)-block injective for every \(m\ge L\). This auxiliary length-shift statement combines the block-injectivity discussion in  [ CPGSV16 , Section II ] with [ CPGSV16 , Appendix C.3, Lemma L ] ; it is not stated there as a separate lemma.