Dependencies

Grouping each consecutive \(p\)-tuple of physical indices defines a canonical computational-basis unitary satisfying

Its forward and inverse coordinate formulas are the corresponding reindexings by the blocked-configuration equivalence and its inverse.

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 \(A^{[L]}\).

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.

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

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 \(G_L(A):=\operatorname{range}(\Gamma _L)\subseteq (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\).

For a tensor \(A\) and a half-chain length \(L\), the half-chain family is the matrix \(\Psi \) with rows indexed by configurations \(\sigma \in \{ 0,\ldots ,d{-}1\} ^L\) and columns indexed by boundary pairs \((\alpha ,\beta )\), whose column at \((\alpha ,\beta )\) is the half-chain vector

The complementary family \(\Phi \) is the matrix with rows indexed by boundary pairs and columns by configurations, whose row at \((\alpha ,\beta )\) is the vector \(|\Psi _{\beta ,\alpha }\rangle \) with the two boundary indices swapped.

The half-chain Gram matrix is \(G=\Psi ^{\dagger }\Psi \), the matrix of overlaps \(G_{(\alpha ,\beta ),(\alpha ',\beta ')} =\langle \Psi _{\alpha ,\beta } | \Psi _{\alpha ',\beta '} \rangle \) of the half-chain vectors.

The coefficient kernel of the ring of \(2L\) sites is the matrix over the two half-chain configuration spaces whose entry at \((\sigma ,\tau )\) is the matrix product vector component \(V^{(2L)}(A)_{\sigma \tau }=\operatorname{tr}(A^{\sigma }A^{\tau })\) of the joined configuration.

The unnormalized half-chain reduced density matrix of the ring state on \(2L\) sites is \(\rho _L=KK^{\dagger }\) with \(K\) the coefficient kernel. Entrywise this is the partial trace of the unnormalized ring state over the complementary half,

and \(\rho _L\) is positive semidefinite.

A finite-dimensional ground space \(S\) is one-dimensional if \(\dim S=1\).

A state \(\psi \) on \(L+1\) sites satisfies the left ground condition if, for every \(j\), the state obtained by fixing the last index to \(j\) lies in \(G_L(A)\).

A state \(\psi \) on \(L+1\) sites satisfies the right ground condition if, for every \(i\), the state obtained by fixing the first index to \(i\) lies in \(G_L(A)\).

A state \(\psi \) on \(K+L\) sites lies in the suffix ground condition if every fixed prefix \(u\) gives a suffix state in \(G_L(A)\).

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 injective \(T\), define the operator \(Q_T:=T S_T T^\dagger \) on its codomain.

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 \(T^\dagger T\).

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

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

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

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

A (translation-invariant, PBC) MPS tensor with physical dimension \(d\) and bond dimension \(D\) is a collection of matrices \(\{ A^i\} _{i=0}^{d-1}\), where \(A^i \in M_{D}(\mathbb {C})\), indexed by a physical index \(i \in \{ 0, \ldots , d{-}1\} \). Such a tensor defines an MPV family. Diagrammatically,

The black node denotes the tensor \(A\), the horizontal legs are virtual, and the upper leg is the physical index \(i\).

The matrix product vector (MPV) of a tensor \(A\) at system size \(N\) is the vector

Equivalently, for a configuration \(\sigma = (i_1, \ldots , i_N) \in \{ 0,\ldots ,d{-}1\} ^N\), we write

The coefficient function \(\sigma \mapsto V^{(N)}(A)_\sigma \) gives the components of the vector in (2); the displayed ket is the corresponding vector in \((\mathbb {C}^d)^{\otimes N}\). For a general word \(w\), we also write \(c_w(A) := \operatorname{tr}(A^w)\). The MPV family generated by \(A\) is the collection \(\mathcal{V}(A) = \bigl\{ |V^{(N)}(A)\rangle \bigr\} _{N \ge 1}\). The coefficient \(V^{(N)}(A)_\sigma \) is the periodic contraction

of \(N\) copies of the local tensor \(A\), with the outer virtual legs closed by the trace.

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.

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 a state \(\psi \) on a periodic chain, define \((T_s\psi )(\sigma )=\psi (T_s\sigma )\).

The chain Hamiltonian is

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

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.

If \(L\le N\) and \(\tau \in \{ 0,\ldots ,d{-}1\} ^{L}\), write \(\sigma ^{[i,i+L-1]\leftarrow \tau }\) for the configuration defined by

The index \(r\) in the first case is unique because \(L\le N\).

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 \(\sigma \mapsto \psi (\sigma _1,\ldots ,\sigma _L,j)\).

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

The transfer map associated to a tensor \(A\) is the linear map \(\mathcal{E}_A : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) defined by

Diagrammatically, the transfer map is the double-layer contraction

in which the upper node denotes \(A\), the lower node denotes \(A^\dagger \), and the physical index is summed over between them.

The word span at length \(n\) is the subspace

This is  [ SPGWC10 , Section II ] .

Let \(A\) be an MPS tensor and let \(L_0,K\ge 0\). Let \(X\) be a matrix, and let \(Y_{c_b}\) be a matrix for each iterated block index \(c_b\) of the complement. Suppose that, for every block letter \(b\) of the alphabet \(\{ 0,\ldots ,d{-}1\} ^{L_0}\) and every iterated block index \(c_b\) of the complement, with \(w(b)\) the length-\(L_0\) word of \(b\) and \(\widetilde{w(c_b)}\) the length-\(L_0K\) complement word obtained by concatenating the blocks of \(c_b\), \(X A^{w(b)} A^{\widetilde{w(c_b)}} = A^{w(b)} Y_{c_b}\). Then there is a family \(Y'_c\), indexed by the length-\(L_0K\) words \(c\), with \(X A^s A^c = A^s Y'_c\) for every length-\(L_0\) word \(s\) and every length-\(L_0K\) word \(c\).

For every \(x\in \ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\), the Euclidean boundary maps intertwine with the blocked-configuration isometry:

Let \(A\) be \(L_0\)-block-injective, with \(L_0{\gt}0\), and let \(L_0\le M\). Suppose two length-\((L_0+1)\) restrictions of a state \(\psi \) at the same cyclic support are represented by \(\operatorname{Res}^\rho _{i,L_0+1}\psi =\Gamma _{L_0+1}(Y_\rho )\) and \(\operatorname{Res}^\tau _{i,L_0+1}\psi =\Gamma _{L_0+1}(Y_\tau )\). If these two restrictions are equal, then \(Y_\rho A^j=Y_\tau A^j\) for every physical letter \(j\). In particular, the last-site boundary-crossing condition and the second boundary-crossing condition satisfy, respectively,

Consequently, for every word \(\sigma \) of length \(L_0\), both implications remain true after right multiplication by \(A^\sigma \).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\). Suppose that matrices \(Y_r(\rho )\), for \(0\le r\le L_0-1\), represent the length-\((L_0+1)\) cyclic restrictions beginning at the sites \(M+1-L_0+r\): \(\operatorname{Res}^\rho _{M+1-L_0+r,L_0+1}\psi =\Gamma _{L_0+1}(Y_r(\rho ))\). Then

For \(L_0=1\), both products are empty.

Let \(A\) be \(L_0\)-block-injective, with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let the matrices \(Y_i(\tau )\) represent its length-\((L_0+1)\) cyclic restrictions, \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a boundary letter \(\eta \) and a complementary word \(\mu \). Let \(\rho ^+\) and \(\rho ^-\) be two boundary conditions satisfying \(\rho ^+_{k+L_0}=\mu _k\) and \(\rho ^-_{k+1}=\mu _k\). If the adjacent-window comparison gives, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\), \(Y_M(\rho ^+)A^jA^\sigma =Y_{M+1-L_0}(\rho ^-)A^jA^\sigma \), then the same product equation holds for \(\tau ^+_\eta (\mu )\) and \(\tau ^-_\eta (\mu )\):

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Let \(X\) be a boundary matrix, and let \(Y_v\) be a matrix for each complementary word \(v\) of length \(K\). Suppose that, for every word \(u\) of length \(L_0\) and every word \(v\) of length \(K\), \(X A^u A^v = A^u Y_v\). Then, for every physical letter \(j\), \(X A^j = A^j X\). This is the boundary-matrix commutation step obtained from the equations \(X A^u A^v=A^uY_v\), for all length-\(L_0\) words \(u\) and all length-\(K\) words \(v\), as in [ CPGSV21 , Section IV.C, lines 2049–2090 ] .

Under the hypotheses of Theorem 12.8.4, one has \(X A^i = A^i X\) for every physical index \(i\).

Let \(L_0{\gt}0\) and \(L_0\le M\), and set

With \(R_j\) as in Lemma 12.9.11,

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Let \(X\) be a boundary matrix, let \(W_\eta \) and \(V_\eta \) be matrix families indexed by a physical letter \(\eta \), and let \(\mu \) be a word. Suppose that, for every physical letter \(j\), \(X A^j = A^jX\). Suppose also that, for every \(\eta \) and every \(j\),

Then, for every \(\eta \) and every \(j\), \(W_\eta A^j = V_\eta A^j\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), and let \(L_0\le M\). Let \(\psi =\Gamma _{M+1}(X)\), and suppose that matrices \(Y_i(\rho )\) represent all length-\((L_0+1)\) cyclic restrictions of \(\psi \). Then, for every boundary condition \(\rho \) and every physical letter \(j\),

For \(\rho =\tau ^-_\eta (\mu )\), this is the transport identity supplied by the adjacent-window argument, with the single factor \(A^j\) following \(Y_{M+1-L_0}(\tau ^-_\eta (\mu ))\). The padded identity, whose corresponding factor is \(A^jA^\sigma \), is supplied separately.

If \(0{\lt}N\), \(L\le N\), and \(\psi \in \mathcal G_{N,L}(A)\), then, for every cyclic window and every outside configuration, there is a boundary matrix \(Y\) such that the corresponding restricted \(L\)-site state is \(\Gamma _L(Y)\).

Let \(A\) be an MPS tensor, let \(L_0{\gt}0\), and let \(L_0\le M\). Suppose the two boundary-crossing restrictions of length \(L_0+1\) are represented by boundary matrices \(Y^+_{\eta ,\mu }\) and \(Y^-_{\eta ,\mu }\):

If, for every physical letter \(j\), \(Y^+_{\eta ,\mu }A^j = Y^-_{\eta ,\mu }A^j\), then \(B^+_{\eta ,\mu }(\psi ) = B^-_{\eta ,\mu }(\psi )\).

Let \(L_0{\gt}0\) and \(L_0\le M\). Let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a complementary word \(\mu \). If, for every boundary letter \(\eta \) and physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\), then, for every \(\eta \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Suppose that, for every outside letter \(\eta \),

Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) such that, for every complementary position \(k\),

and, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a complementary word \(\mu \) and a matrix \(X\). Suppose that, for every boundary letter \(\eta \) and every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j=A^\mu A^jX\). Suppose also that, for every \(\eta \), \(j\), and every pair of words \(\sigma ,\alpha \) of length \(L_0\),

Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfying the complementary-word equations

and the product equation

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a complementary word \(\mu \) and a matrix \(X\). Suppose that, for every boundary letter \(\eta \) and every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = A^\mu A^jX\). Suppose also that, for every \(\eta \), \(j\), and every word \(\sigma \) of length \(L_0\),

Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) such that, for every complementary position \(k\),

and, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a complementary word \(\mu \) of length \(M+1-(L_0+1)\). Suppose that, for every boundary letter \(\eta \) and every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\). Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) such that, for every complementary position \(k\),

and, for every physical letter \(j\) and every word \(\sigma \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) and assume that every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\). Then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every word \(\alpha \) of length \(L_0\), \(X A^\alpha A^\nu = A^\alpha Y_\nu \).

Let \(A\) be a tensor with virtual dimension at least one whose single-site matrices span the full matrix algebra. Let \(L_0{\gt}0\) and \(L_0{\lt}M\), and let \(\psi =\Gamma _{M+1}(X)\). If every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\), then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every word \(\alpha \) of length \(L_0\), \(X A^\alpha A^\nu = A^\alpha Y_\nu \).

Let \(A\) be a tensor with virtual dimension at least one, and let \(L_0{\gt}0\) and \(L_0{\lt}M\). Let \(\psi =\Gamma _{M+1}(X)\) and assume that every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\). Then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every physical letter \(j\) and every word \(\alpha \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and suppose matrices \(Y_i(\rho )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\rho _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\rho ))\). Then, for every boundary condition \(\rho \),

Each side has \(M+2-L_0\) one-site factors, with site labels read modulo \(M+1\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and suppose matrices \(Y_i(\rho )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\rho _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\rho ))\). Then, for every boundary condition \(\rho \),

For \(L_0=1\), both products are empty. The same equation remains true after multiplying both sides on the right by any matrix \(R\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(L_0\le M\). Suppose \(\psi =\Gamma _{M+1}(X)\), and suppose matrices \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Then, for every physical letter \(j\) and boundary condition \(\tau \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(L_0\le M\). Let \(\psi \) be a vector on \(M+1\) sites, and suppose the two length-\((L_0+1)\) boundary restrictions are represented by \(\Gamma _{L_0+1}(Y_M(\tau ^+_\eta (\mu )))\) and \(\Gamma _{L_0+1}(Y_{M+1-L_0}(\tau ^-_\eta (\mu )))\). If these two restrictions are equal, then for every physical letter \(j\), \(Y_M(\tau ^+_\eta (\mu ))A^j = Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j\). Thus the first-letter restriction of the preceding equality is precisely the displayed product equation.

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(L_0\le M\). Suppose \(\psi =\Gamma _{M+1}(X)\), and suppose matrices \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). For every boundary letter \(\eta \), every physical letter \(j\), and every complementary word \(\mu \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), let \(L_0{\lt}M\), and let

For every boundary letter \(\eta \) and complementary word \(\mu \), the two restrictions crossing the periodic cut satisfy

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), let \(L_0{\lt}M\), and let \(\psi \in \mathcal G_{M+1,L_0+1}(A)\) with \(\psi =\Gamma _{M+1}(X)\). Let \(R_j\) denote restriction to the first physical letter \(j\). For every outside letter \(\eta \), complementary word \(\mu \), and physical letter \(j\), one has

This is the one-letter product form of the local comparison corresponding to [ CPGSV21 , lines 2078–2079 ] .

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Fix a complementary word \(\mu \), a matrix \(X\), and a family of matrices \(Y_i(\tau )\). Suppose that, for every boundary letter \(\eta \), every physical letter \(j\), and every pair of words \(\sigma ,\alpha \) of length \(L_0\),

Then, for every \(\eta \), \(j\), and \(\sigma \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), and let \(L_0\le M\). Fix a complementary word \(\mu \), a matrix \(X\), and a family of matrices \(Y_i(\tau )\). Suppose that, for every boundary letter \(\eta \), every physical letter \(j\), and every word \(\sigma \) of length \(L_0\),

Then, for every \(\eta \) and \(j\), \(Y_{M+1-L_0}(\tau ^-_\eta (\mu ))A^j = A^\mu A^jX\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(L_0\le M\). Let \(\psi \) be a vector on \(M+1\) sites, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Assume that for every boundary letter \(\eta \),

Then for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),

Suppose a single family of boundary matrices \(Y_\mu \), indexed by words \(\mu \) on the complementary sites, satisfies the two one-sided identities

for every pair of physical letters \(a,b\). Then \(X\) commutes with every word product \(A^aA^\mu A^b\) obtained by adjoining one letter on each side of the word \(A^\mu \).

If an \(N\)-site state satisfies the \(L\)-site ground condition on every non-wrapping contiguous window, with \(L \ge 2\), then it lies in \(G_N(A)\).

On a nonempty periodic chain, longer cyclic-window constraints imply all shorter cyclic-window constraints: if \(L'\le L\le N\), then \(\mathcal G_{N,L}(A)\subseteq \mathcal G_{N,L'}(A)\).

If \(w = (J_1, \ldots , J_m)\) is a word in the blocked alphabet, where each \(J_k\) is an \(L\)-tuple in \(\{ 0, \ldots , d{-}1\} ^L\), then concatenating the block words gives a word \(\widetilde{w}\) of length \(mL\) in the original alphabet and \((A^{[L]})^w = A^{\widetilde{w}}\).

If \(L\le N\), then restricting to the same cyclic interval after prescribing its values recovers the inserted word: \(\left(\sigma ^{[i,i+L-1]\leftarrow \tau }\right)|_{[i,i+L-1]}=\tau \).

Let \(R_j\) be restriction to first letter \(j\):

For \(\phi ,\psi \in (\mathbb {C}^d)^{\otimes (L+1)}\),

For every \(L\), one has \(\dim G_L(A)\le D^2\).

If \(\psi \in G_{L+1}(A)\), i.e. if \(\psi (\sigma )=\operatorname{tr}(A^\sigma X)\) for some \(X\), then, for each \(j\), the state obtained by fixing the last index to \(j\) lies in \(G_L(A)\) with witness \(A^jX\).

If \(\psi \in G_{L+1}(A)\) with \(\psi (\sigma )=\operatorname{tr}(A^\sigma X)\), then, for each \(i\), the state obtained by fixing the first index to \(i\) lies in \(G_L(A)\) with witness \(XA^i\).

If \(\psi \in G_{K+L}(A)\), then every fixed-prefix suffix restriction of \(\psi \) lies in \(G_L(A)\). For a vector in \(G_{L+1}(A)\), fixing either endpoint gives the expected left or right multiplication of its boundary matrix.

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

If \(d^L{\gt}D^2\), then \(G_L(A)\) is a proper subspace of the local Hilbert space, namely \(G_L(A)\neq (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\).

For a diagonal fixed point \(\Lambda =\operatorname {diag}(\lambda _1,\ldots ,\lambda _D)\) with \(\operatorname{tr}(\Lambda )\neq 0\), the limiting overlaps form a diagonal matrix in the boundary pairs:

For \(\operatorname{tr}(\Lambda )=1\) this is the display \(\lambda _\alpha \delta _{\alpha ,\alpha '}\delta _{\beta ,\beta '}\) of [ PGVWC07 ] .

The half-chain overlaps are matrix elements of the \(L\)-fold transfer map on matrix units:

The coefficient kernel factors as the product \(\Psi \Phi \) of the half-chain family with its complementary family. Equivalently,

Assume \(\psi \) on \(K+L_0+1\) sites satisfies the left ground condition on the first \(K+L_0\) sites and the suffix ground condition on every fixed prefix of length \(K\). Then there exist matrices \((Z_j)_j\) and \((Y_u)_u\) with the two trace representations below and, for all \(j\) and \(u\), the compatibility relation

For each site \(i\), one has \(h_iV^{(N)}(A)=0\).

If \(L\le N\), then the translated term is obtained by applying the local parent interaction on the chosen cyclic window:

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

Let \(A\) be an \(e\times d\) matrix and \(B\) a \(d\times e\) matrix, with \(d\leq e\). Then

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

Assume \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\) and \(M\ge L_0\). If the second boundary-crossing reduced cyclic interval has boundary matrices \(Y_\tau \), then \(X A^j C^-_\tau =A^jY_\tau \) for every physical letter \(j\), where \(C^-_\tau \) is the complementary word seen from the second cyclic position.

For every chain length \(N\), one has \(V^{(N)}(A)=\Gamma _N(\mathbb {1})\).

If \(0{\lt}N\) and \(L\le N\), then \(V^{(N)}(A)\in \mathcal G_{N,L}(A)\).

\(V^{(N)}(A)\in G_N(A)\) for every \(N\), with boundary matrix \(X=\mathbb {1}\).

For any window of \(L\) consecutive sites starting at position \(i\) in an \(N\)-site periodic chain with \(L\le N\), the restriction of the periodic MPS vector \(V^{(N)}(A)\) to that window lies in \(G_L(A)\).

The above coefficient space satisfies \(\dim (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})=d^L\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). If two boundary matrices have the same product with every one-site tensor on one fixed side, then the two boundary matrices are equal.

Let \(S_m\subseteq (\mathbb {C}^d)^{\otimes m}\) be subspaces, and let \(0{\lt}L\le N\). Suppose that every non-wrapping length-\(L\) interval of \(\psi \), for every fixed choice of the complementary sites, lies in \(S_L\). Suppose also that, for every \(m\ge L\), \((\mathbb {C}^d\otimes S_m)\cap (S_m\otimes \mathbb {C}^d) = S_{m+1}\). Then \(\psi \in S_N\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(L_0\le M\). If an outside configuration \(\rho \) has the same word \(\mu \) as \(\tau ^+_\eta (\mu )\) on the sites outside the last-site cyclic window, then the two matrices representing that restriction agree: \(Y_\rho = Y_{\tau ^+_\eta (\mu )}\). The same assertion holds for the second boundary-crossing window: if \(\rho \) has the same outside word as \(\tau ^-_\eta (\mu )\), then \(Y_\rho = Y_{\tau ^-_\eta (\mu )}\). This is the boundary-matrix independence step used to choose one representative outside configuration for each outside word in the closure-property matrix comparison.

If \(A\) is \(L_0\)-block-injective, \(q \ge 1\), and \(k \le qL_0\), then the two one-sided annihilation statements are

The second implication is the left-handed form.

If \(\beta \) has length \(K\) and \(\alpha \) has positive length \(S\), then \(T_K(\beta \alpha )=\alpha \beta \) as configurations on the periodic chain of length \(K+S\).

If \(\psi \in \mathcal G_{N,L}(A)\), then \(T_s\psi \in \mathcal G_{N,L}(A)\) for every cyclic translation \(s\).

For every \(N\) with \(L\le N\), the periodic MPS vector satisfies \(H_N(A,L)V^{(N)}(A)=0\).

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

For any \(v\in G_L(A)\), one has \(h_L(A)v=0\).

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

Let \(S_m\subseteq (\mathbb {C}^d)^{\otimes m}\) be subspaces, and let \(0{\lt}L\le N\). Suppose that \(G_L(A)\subseteq S_L\). Suppose also that, for every \(m\ge L\), \((\mathbb {C}^d\otimes S_m)\cap (S_m\otimes \mathbb {C}^d) = S_{m+1}\). Then the periodic local constraints imply \(\mathcal G_{N,L}(A)\subseteq S_N\).

If \(L\le N\), then prescribing on a cyclic interval the values already carried by \(\sigma \) leaves \(\sigma \) unchanged: \(\sigma ^{[i,i+L-1]\leftarrow \sigma |_{[i,i+L-1]}}=\sigma \).

If \(L\le N\), then two prescriptions on the same cyclic interval reduce to the second prescription:

Fix a physical letter \(\eta \) used outside the complementary sites. Suppose the two one-sided boundary matrix families \(Y^+\) and \(Y^-\) have been extracted from the cyclic windows. If, for every word \(\mu \) on the complementary sites, these boundary matrices agree on the boundary conditions built from \(\eta \), \(Y^+_{\tau ^+_\eta (\mu )}=Y^-_{\tau ^-_\eta (\mu )}\), then there is a single family \(Y_\mu \) satisfying

for all letters \(a,b\).

If a boundary matrix \(X\) commutes with every word product of length \(m\), then it commutes with every word product whose length is a multiple \(q\, m\).

One has \(S_L(A)=M_{D}(\mathbb {C})\) if and only if \(A\) is \(L\)-block injective.

On a chain of length \(N=M+1\), with \(L_0\le M\), fix a physical letter \(\eta \) used outside the complementary sites and a word \(\mu \) of length \(M+1-(L_0+1)\). Write \(\tau ^+_\eta (\mu )\) for the boundary condition at the support crossing the last site and \(\tau ^-_\eta (\mu )\) for the boundary condition at the second boundary-crossing support. Their exposed complements are both exactly \(\mu \).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). Fix two matrix families \(Y^+\) and \(Y^-\) and the two reindexed boundary conditions \(\tau ^+_\eta (\mu )\) and \(\tau ^-_\eta (\mu )\). If, for every physical letter \(j\), \(Y^+_{\tau ^+_\eta (\mu )} A^j = Y^-_{\tau ^-_\eta (\mu )} A^j\), then \(Y^+_{\tau ^+_\eta (\mu )} = Y^-_{\tau ^-_\eta (\mu )}\).

Assume \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\) and \(M\ge L_0\). If the reduced cyclic interval crossing the last site has boundary matrices \(Y_\tau \), then, for every physical letter \(j\), one has

Here \(C^+_\tau \) is the complementary word seen by that cyclic interval.

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 \(K_0\) be invertible in a normed ring with convergent geometric series and \(\lVert I\rVert =1\). If \(a{\lt}1\) and

then \(K\) is invertible and

The same conclusion follows from \(\lVert K_0^{-1}\rVert \, \lVert K-K_0\rVert \le a\).

Let \(K_0\) be invertible in a normed ring with convergent geometric series and \(\lVert I\rVert =1\). If \(a{\lt}1\) and (30) holds, then

The same estimate follows from \(\lVert K_0^{-1}\rVert \, \lVert K-K_0\rVert \le a\).

Let \(A\) be \(L_0\)-block-injective. Let \(X\) be a boundary matrix, and let \(Y_\nu \) be a matrix for each complementary word \(\nu \) of length \(K\). Suppose that, for every word \(\alpha \) of length \(L_0\), every word \(\beta \) of length \(L_0\), and every complementary word \(\nu \),

Then, for every word \(\alpha \) of length \(L_0\) and every complementary word \(\nu \), \(X A^\alpha A^\nu = A^\alpha Y_\nu \).

If \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\), \(m \ge L_0\), and \(X \in M_{D}(\mathbb {C})\) commutes with every length-\(m\) word product, then \(X\) commutes with every matrix in \(M_{D}(\mathbb {C})\).

If \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\), \(m \ge L_0\), and \(X \in M_{D}(\mathbb {C})\) commutes with every length-\(m\) product \(A^\omega \), then \(X\) already commutes with every length-\(L_0\) product \(A^u\).

Assume \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\). Let \(\{ Z_u\} _{|u|=L_0}\) be a family indexed by words of length \(L_0\). If, for some \(K \ge 0\) and every suffix word \(w\) of length \(K\), there exists \(Y_w \in M_{D}(\mathbb {C})\) such that \(Z_u A^w = A^u Y_w\) for every word \(u\) of length \(L_0\), then there exists \(X \in M_{D}(\mathbb {C})\) with \(Z_u = A^u X\) for every word \(u\) of length \(L_0\).

The Euclidean local space of the blocked tensor is carried exactly to the original Euclidean local space:

Hence their orthogonal projectors satisfy

These statements include \(N=0\) or \(p=0\); in those cases the same explicit configuration isometry and boundary-map calculation apply. In particular, the projector identity holds for three blocked sites and \(3p\) original sites. These identities concern the full local ground spaces and their orthogonal projectors under blocking.

Let \(A\) be \(L_0\)-block-injective, with \(L_0{\gt}0\), and let \(L_0\le M\). Let \(\psi \) be a state on the chain of length \(M+1\), and let the matrices \(Y_i(\tau )\) represent its length-\((L_0+1)\) cyclic restrictions, \(\operatorname{Res}^\tau _{i,L_0+1}(\psi )=\Gamma _{L_0+1}(Y_i(\tau ))\). Fix a boundary letter \(\eta \) and a complementary word \(\mu \). Suppose that, for each physical letter \(j\) and word \(\sigma \) of length \(L_0\), boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfy, for every complementary position \(k\),

If, for every \(j\) and \(\sigma \),

then, for every \(j\) and \(\sigma \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Let \(\psi \in \mathcal G_{M+1,L_0+1}(A)\) have an open-chain representation \(\psi =\Gamma _{M+1}(X)\). Let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi ) = \Gamma _{L_0+1}(Y_i(\tau ))\). If \(L_0{\lt}M\), then for every letter \(\eta \) used in the two boundary conditions, every word \(\mu \) on the complementary sites, and every physical letter \(j\),

This is the one-site product equality used to compare the two boundary matrices in the periodic-boundary comparison.

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Let \(\psi \in \mathcal G_{M+1,L_0+1}(A)\) have an open-chain representation \(\psi =\Gamma _{M+1}(X)\). Let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi ) = \Gamma _{L_0+1}(Y_i(\tau ))\). If \(L_0{\lt}M\), then for every boundary letter \(\eta \), complementary word \(\mu \), physical letter \(j\), and word \(\sigma \) of length \(L_0\),

Let \(A\) be injective with \(D\ge 1\), and let \(\psi =\Gamma _N(X)\) for some boundary matrix \(X\in M_{D}(\mathbb {C})\). If every cyclic restriction of length \(L\) of \(\psi \) lies in \(G_L(A)\), with \(N\ge 2\) and \(1{\lt}L\le N\), then \(X A^j=A^jX\) for every \(j=0,\ldots ,d-1\).

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

If \(A\) is normal, \(D \ge 1\), and injective after blocking \(L_0{\gt}0\) sites, with \(N \ge 2\), \(L_0+1\le N\), and \(L_0 {\lt} L \le N\), then \(\mathcal G_{N,L}(A) = \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Let \(\psi \) be a state on the chain of length \(M+1\) with open-chain representation \(\psi =\Gamma _{M+1}(X)\). Let \(Y_i(\tau )\) be matrices satisfying \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi ) = \Gamma _{L_0+1}(Y_i(\tau ))\). If \(L_0{\lt}M\), then for every nonempty complementary word \(\mu \) there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\), depending on the physical letter \(j\) and the word \(\sigma \) of length \(L_0\), such that, for every complementary position \(k\),

and, for every \(j\) and \(\sigma \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a complementary word \(\mu \). Suppose that, for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\sigma ,\alpha \),

Then there are boundary conditions \(\rho ^+_{j,\sigma }\) and \(\rho ^-_{j,\sigma }\) satisfying the complementary-word equations

and the product equation

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) and assume that every cyclic restriction of length \(L_0+1\) belongs to \(G_{L_0+1}(A)\). Then there are boundary matrices \(Y_\nu \), indexed by nonempty complementary words \(\nu \) of length \(M+1-(L_0+1)\), such that, for every word \(\alpha \) of length \(L_0\) and every word \(\beta \) of length \(L_0\),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation. Assume that, for all \(i\) and \(\tau \), \(\operatorname{Res}^{\tau }_{i,L_0+1}(\psi )\in G_{L_0+1}(A)\). In local coordinate notation, fix a nonempty word \(\mu \) on the sites outside the two cyclic windows. For every outside letter \(\eta \), choose the two outside configurations so that, for each index \(k\) of this outside word,

and put the letter \(\eta \) on the remaining sites. The notation \(\tau ^\pm _\eta (\mu )\) is a coordinate parametrization for the comparison used at the periodic boundary in [ CPGSV21 , Section IV.C, lines 2078–2079 ] ; it is not notation from the source. The corresponding boundary-crossing restriction equality is

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0\le M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a complementary word \(\mu \). Suppose that, for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),

Then, for every boundary letter \(\eta \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a nonempty complementary word \(\mu \). Then for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\), \(L_0{\lt}M\), and \(D\ge 1\). Let \(\psi =\Gamma _{M+1}(X)\) be an open-chain representation, and let \(Y_i(\tau )\) represent the length-\((L_0+1)\) cyclic restrictions of \(\psi \). Fix a nonempty complementary word \(\mu \). Then for every boundary letter \(\eta \), physical letter \(j\), and length-\(L_0\) words \(\alpha ,\sigma \),

Let \(A\) be injective after blocking \(L_0{\gt}0\) sites and let \(D \ge 1\). Assume \(N \ge 2\) and \(L_0 {\lt} L \le N\). Suppose that, for every physical letter \(\eta \) used in the two boundary conditions and every boundary matrix obtained from an \(N\)-site chain ground state, the identities

hold for every physical letter \(j\) and every complementary word \(\mu \), and are supplemented, for every \(\mu \), by \(Y^+_{\tau ^+_\eta (\mu )} = Y^-_{\tau ^-_\eta (\mu )}\). Then \(\mathcal G_{N,L}(A)\subseteq \operatorname{span}\{ V^{(N)}(A)\} \).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D \ge 1\). If an \(N\)-site periodic-chain state satisfies all cyclic ground-space constraints at some range \(L_0+1\le L\le N\), then it lies in the open-chain ground space \(G_N(A)\).

If a cyclic \((L+1)\)-window is represented by a boundary matrix \(Y\), then deleting one endpoint gives

Hence two adjacent cyclic windows whose restrictions agree on the common length-\(L\) overlap have boundary matrices satisfying \(Y_1A^a=A^bY_2\). Consequently, for a finite chain of adjacent overlaps indexed by \(0\le r{\lt}n\), satisfying \(Y_rA^{a_r}=A^{b_r}Y_{r+1}\), one has \(Y_0A^{a_0}\cdots A^{a_{n-1}} = A^{b_0}\cdots A^{b_{n-1}}Y_n\). In particular, if the two endpoint words are named by the equations \(a_r=\rho (i_0+r)\) and \(b_r=\rho (i_0+r+L+1)\), with site labels read modulo \(N\), then the same transport identity is written as \(Y_0A^a=A^bY_n\).

Let \(I\) be a finite index set and let \(B=(B_{ij})_{i,j\in I}\) be a complex matrix. Then the squared operator norm of \(B\) on \(\ell ^2(I)\) satisfies

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 \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), and let \(\psi \in \mathcal G_{L_0+1,L_0+1}(A)\) satisfy \(\psi =\Gamma _{L_0+1}(X)\). Then \(XA^j=A^jX\) for every physical letter \(j\).

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\). If

then \(XA^j=A^jY\) for every physical letter \(j\).

If \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), then

Let \(V\) be a complex Banach space and let \(T\colon V\to V\) be a bounded linear operator. If \(\rho _{\mathrm{spec}}(T){\lt}1\), then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that

for every \(n\in \mathbb {N}\) and \(x\in V\).

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

For every complex-linear map \(\mathcal L\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), the sum of squared entry norms of \(R(\mathcal L)\) equals that of the rectangular Choi matrix:

The two matrices differ only by a reshuffling of indices; the equality is entrywise Frobenius, not operator norm.

Let \(I\) be a finite index set, equip \(\mathbb C^{I\times I}\) with its \(L^2\) operator norm, and let \(\mathcal L\colon \mathbb C^{I\times I}\to \mathbb C^{I\times I}\) be complex-linear. Denote the operator norm of \(\mathcal L\) for these matrix norms by \(\lVert \mathcal L\rVert _{\mathrm{op},L^2}\). Then

Let \(I\) be a finite index set and let \(\mathcal L\colon \mathbb C^{I\times I}\to \mathbb C^{I\times I}\) be complex-linear. Then

Let \(\mathcal L\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be complex-linear and let \(\mathbf e_{(j,i)}\) be the standard unit vector at \((j,i)\). For all \(a,b,c,e\in \{ 0,\ldots ,D{-}1\} \),

This is the coordinate identity in (5); it is not a Hilbert–Schmidt pairing of \(E_{ab}\) with \(\mathcal L(E_{ce})\).

For complex-linear maps \(\mathcal L,\mathcal M\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\),

Suppose that \(L_0{\gt}0\), that \(A\) is \(L_0\)-block injective, and that \(L\ge L_0\). Then the physical orthogonal projector onto the local ground space is

Thus the displayed operator is the physical range projector on \(\ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\).

Assume \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\). If an element \(\psi \) on \(K+L_0+1\) sites satisfies the left ground condition on the first \(K+L_0\) sites and the suffix ground condition on every prefix of length \(K\), then \(\psi \in G_{K+L_0+1}(A)\).

If \(A\) is injective and \(L \ge 1\), then \(\dim G_L(A) = D^2\).

Let \(\mathbf e_{(j,i)}:=\operatorname {single}_{(j,i)}(1)\) denote the standard unit vector in \(\ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\), and let \(\mathcal E_A\) be the transfer map. Then

Under Frobenius vectorization, \(\mathbf e_{(b,a)}=U(E_{ab})\) and \(\mathbf e_{(e,c)}=U(E_{ce})\), explaining the reversed coordinate order. Thus the Gram operator is the indicated Choi reshuffling; in particular it is not obtained by replacing the right-hand side with \(\langle E_{ab},\mathcal E_A^L(E_{ce})\rangle _{\mathrm{HS}}\).

The ground-space Gram operator is exactly the operator-level reshuffling of the Choi matrix of the iterated transfer map:

The equality is an equality of operators on \(\ell ^2(\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} )\). Its entries are those in (7), rather than Hilbert–Schmidt pairings with \(\mathcal E_A^L\).

Let \(A\) be an MPS tensor of bond dimension \(D\), let \(\rho \in M_{D}(\mathbb {C})\) satisfy \(\operatorname{tr}(\rho )\neq 0\), and let \(P_\rho \) be the fixed-point projection

Suppose that \(\mathcal E_A\) is trace-preserving, \(\mathcal E_A(\rho )=\rho \), and

Then there are real constants \(C,r\) with \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that, for every \(n\geq 1\),

No positivity, injectivity, or uniqueness hypothesis is imposed.

Let \(A\) be an MPS tensor of bond dimension \(D\), let \(\rho \in M_{D}(\mathbb {C})\) satisfy \(\operatorname{tr}(\rho )\neq 0\), and let \(P_\rho \) be the corresponding fixed-point projection. If \(\mathcal E_A\) is trace-preserving, \(\mathcal E_A(\rho )=\rho \), and \(\rho _{\mathrm{spec}}(\mathcal E_A-P_\rho ){\lt}1\), then

in operator norm as \(n\to \infty \).

For injective \(A\) and \(L \ge 2\), \(G_L^{\mathrm{left}}\cap G_L^{\mathrm{right}} = G_{L+1}(A)\), equivalently, \(\psi \in G_{L+1}(A)\) if and only if \(\psi \in G_L^{\mathrm{left}}\cap G_L^{\mathrm{right}}\).

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 \(A\) is \(L\)-block injective, then \(\Gamma _L^{\mathrm{ES}}\) is injective.

Suppose that \(L_0{\gt}0\), that \(A\) is \(L_0\)-block injective, and that \(L\ge L_0\). Then \(\Gamma _L^{\mathrm{ES}}\) is injective.

If \(A\) is \(L\)-block injective, then the injective range projector of \(\Gamma _L^{\mathrm{ES}}\) is the physical orthogonal projector onto the local ground space:

In particular, this is the physical range projector on the local Hilbert space, not the virtual transfer fixed-point projector.

Let \(e_{(b,a)}\) denote the Frobenius coordinate vector representing the virtual matrix unit \(E_{ab}\). Then \(\Gamma _L^{\mathrm{ES}}(e_{(b,a)})=\iota _L(\Gamma _L(E_{ab}))\).

If \(A\) is injective and \(L \ge 1\), then \(\Gamma _L\) is injective.

If \(A\) is \(L_0\)-block injective, then \(\Gamma _{L_0}\) is injective.

If the length-\(L\) products \(\{ A^\sigma : |\sigma | = L\} \) span \(M_{D}(\mathbb {C})\), then \(\Gamma _L\) is injective.

Let \(A\) be injective after blocking \(L_0{\gt}0\) sites. Suppose there are matrices \(Y_\mu \) such that

for every pair of physical letters \(a,b\). Then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).

Let \(A\) be injective after blocking \(L_0{\gt}0\) sites, and let \(m \ge L_0\). If a boundary matrix \(X\) commutes with every length-\(m\) word product, then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).

Let \(A\) be injective after blocking \(L_0{\gt}0\) sites. If \(X\) commutes with every word product of some positive length \(m\), then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).

Let \(A\) be injective after blocking \(L_0{\gt}0\) sites, and fix a physical letter \(\eta \) used outside the complementary sites. Suppose that, for every boundary condition \(\tau \) and every physical letter \(j\),

where \(C^+_\tau \) and \(C^-_\tau \) are the complementary words exposed by the two boundary-crossing cyclic windows. Suppose also that the two boundary matrices satisfy \(Y^+_{\tau ^+_\eta (\mu )} = Y^-_{\tau ^-_\eta (\mu )}\), for every word \(\mu \) on the complementary sites. Then \(\Gamma _N(X)\in \operatorname{span}\{ V^{(N)}(A)\} \).

For an injective tensor \(A\) with \(D \ge 1\), if \(N \ge 2\) and \(1 {\lt} L \le N\), then the periodic chain ground space is one-dimensional.

The characteristic polynomials of the half-chain reduced density matrix and of the boundary comparison matrix \((SG^{\mathsf T}S)G\) of size \(D^2\) agree after matching the trivial kernel factors:

In particular the nonzero spectrum of \(\rho _L\), with multiplicities, is that of the boundary comparison matrix.

Let \(A\) have trace-preserving transfer map with diagonal fixed point \(\Lambda =\operatorname {diag}(\lambda _1,\ldots ,\lambda _D)\), \(\operatorname{tr}(\Lambda )\neq 0\), satisfying the spectral condition (98). Then there are constants \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that for every \(L\geq 1\) the boundary comparison matrix satisfies, entrywise,

The limiting matrix is the diagonal matrix \(\Lambda \otimes \Lambda \) up to the trace normalization. Together with Theorem 12.10.8, the nonzero spectrum of the half-chain reduced density matrix on the ring of \(2L\) sites is that of a \(D^2\times D^2\) matrix converging entrywise, geometrically in \(L\), to \(\Lambda \otimes \Lambda /\operatorname{tr}(\Lambda )^2\). The remaining passage to the convergence of the eigenvalues themselves is the continuity of the spectrum of a matrix in its entries and is not part of this section.

Let \(A\) have trace-preserving transfer map \(\mathcal{E}_A\) with fixed point \(\Lambda \), \(\operatorname{tr}(\Lambda )\neq 0\), and suppose

where \(P_{\Lambda }\) is the fixed-point projection. Then there are constants \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that for every \(L\geq 1\) and all boundary pairs,

The hypothesis (98) expresses the generic condition C2 of [ PGVWC07 ] in the trace-preserving gauge: apart from the simple eigenvalue one, the spectrum of \(\mathcal{E}_A\) lies strictly inside the unit disc. The proof chooses \(r\) strictly between the spectral radius of \(\mathcal{E}_A-P_{\Lambda }\) and one. Without a semisimplicity assumption on the subleading spectrum, the theorem does not assert the same estimate with \(r\) equal to that spectral radius.

Let \(S\) denote the swap of the two boundary indices. The unnormalized half-chain reduced density matrix factors exactly as

where \(SG^{\mathsf T}S\) is the Gram matrix of the complementary family.

For finite-dimensional \(S\), \(\dim S=1\) if and only if there exists a nonzero \(\psi _0\in S\) such that every \(\psi \in S\) is of the form \(c\, \psi _0\).

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

Let \(\mathbb K\) be a real or complex scalar field, let \(E\) be a finite-dimensional complete \(\mathbb K\)-Hilbert space, and let \(F\) be a complete \(\mathbb K\)-Hilbert space. Let \(T\colon E\to F\) be an injective continuous linear map. If \(a\in \mathbb R\) satisfies \(a{\lt}1\) and

then

Let \(K_i\) be a net in a normed ring with convergent geometric series. If \(K_i\to K_0\) and \(K_0\) is invertible, then

Let \(I\) be a finite index set. For every \(i,j\in I\), the matrix unit \(E_{ij}\) satisfies

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 \(G_L(A)\).

For every \(X\in M_{D}(\mathbb {C})\) and \(N\geq 0\),

where \(A^\sigma =A^{\sigma _1}\cdots A^{\sigma _N}\) denotes the word evaluation of Definition 2.1.2.

If \(A\) is \(L_0\)-block-injective with \(L_0{\gt}0\), \(D\ge 1\), and \(N\ge L_0+1\), then \(V^{(N)}(A)\ne 0\) in the \(N\)-site space.

If \(\lVert t\rVert \le a{\lt}1\) in a normed ring with convergent geometric series and \(\lVert 1\rVert =1\), then

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 \(A\) is normal, \(D \ge 1\), and injective after blocking \(L_0{\gt}0\) sites, with \(N \ge 2\), \(L_0+1\le N\), and \(L_0 {\lt} L \le N\), then \(\mathcal G_{N,L}(A) \subseteq \operatorname{span}\bigl\{ V^{(N)}(A)\bigr\} \).

If \(D\ge 1\), \(A\) 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 offsets \(s,t\in \mathbb Z/N\mathbb Z\) and a state \(\psi \) on the periodic chain,

If \(D \ge 1\), \(A\) is injective after blocking \(L_0{\gt}0\) sites, and \(L_0 {\lt} L \le N\), then the parent Hamiltonian of interaction range \(L\) has a unique periodic ground state. These inequalities already imply \(N\ge 2\) and \(N\ge L_0+1\). Thus this strengthens the finite-size range \(N\ge 2L_0\) printed in PGVWC07, Theorem uniqueGS  [ PGVWC07 ] .

Let \(A\) be a primitive MPS tensor whose fixed-point witness \(\rho \) is positive definite, and put \(\mathcal K_\infty =\mathscr R(P_\rho )\). For every \(0{\lt}a{\lt}1\), all sufficiently large \(n\) satisfy

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 \(n\geq 1\),

Let \(A\) be a primitive MPS tensor of nonzero bond dimension whose fixed-point witness \(\rho \) is positive definite. Put

There are constants \(g,c,r{\gt}0\) with \(r{\lt}1\) and \(c=\lVert \mathcal I_\infty \rVert g\) such that, for every \(n\geq 1\),

Whenever \(cr^n{\lt}1\), the operator \(\mathcal K_n\) is invertible and

Under the same primitive-MPS and positive-definite fixed-point hypotheses,

in operator norm.

If \(A\) is a primitive MPS tensor with fixed-point witness \(\rho \), then

in operator norm as \(n\to \infty \).

Under the same hypotheses, for every \(0{\lt}a{\lt}1\) and all sufficiently large \(n\), the physical boundary map \(\Gamma _n^{\mathrm{ES}}\) is injective and its inverse Gram operator is \(\mathcal K_n^{-1}\). Moreover,

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

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

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\). Suppose \(\psi \in \mathcal G_{N,L}(A)\), \(N\ge 2\), \(L_0{\lt}L\le N\), and \(\psi =\Gamma _N(X)\) for a boundary matrix \(X\). Then there are two families of boundary matrices \(Y^+_\tau \) and \(Y^-_\tau \), indexed by fixed complement values \(\tau \), such that, for every physical letter \(j\),

Here \(C^+_\tau \) and \(C^-_\tau \) are the complementary words exposed by the two boundary-crossing cyclic intervals, both of reduced length \(L_0+1\).

Assume \(A\) is \(L_0\)-block-injective with \(L_0 {\gt} 0\). If matrices \((Z_j)_j\) satisfy \(Z_jA^\sigma =A^jY_\sigma \) for every word \(\sigma \) of length \(K\), then there exists a matrix \(X\) such that \(Z_j=A^jX\) for all \(j\).

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

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

If \(A\) is injective, then \(\ker \Phi _A = \{ 0\} \).

If \(A\) becomes injective after blocking \(L_0 {\gt} 0\) sites and \(D \ge 1\), the parent Hamiltonian with interaction range \(2L_0\) has a unique ground state on every periodic chain with \(N \ge 2L_0\). In particular, this includes \(L_0=1\) and \(N=2\).

If \(A\) is normal, becomes injective after blocking \(L_0 {\gt} 0\) sites, and \(D \ge 1\), then the parent Hamiltonian with interaction range \(L_0 + 1\) has a unique ground state on every periodic chain with \(L_0+1\le N\).

Let \(A\) be a tensor whose length-\(K\) word products span \(M_{D}(\mathbb {C})\). Let \(\{ F_b\} _{b\in B}\) and \(\{ Z_b\} _{b\in B}\) be two families of matrices. If, for every length-\(K\) word \(w\), there is a matrix \(Y_w\) such that \(Z_b A^w = F_bY_w\) for every \(b\in B\), then there is a single matrix \(Y\) such that \(Z_b = F_bY\) for every \(b\in B\).

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.

Let \(A\) be \(L_0\)-block-injective with \(L_0{\gt}0\) and \(D\ge 1\), let \(N\ge 2\), \(L_0+1{\lt}N\), and \(L_0{\lt}L\le N\), and let \(\psi \in \mathcal G_{N,L}(A)\) have an open-chain representation \(\psi =\Gamma _N(X)\). Suppose \(Y^{+}\) and \(Y^{-}\) are two families of matrices obtained from the two boundary-crossing cyclic-window constraints, and that they satisfy, for every physical letter \(j\) and every boundary condition \(\tau \),

For every physical letter \(\eta \in \{ 0,\ldots ,d{-}1\} \) and every word \(\mu \) on the complementary sites, define two boundary conditions by

Then the two families agree on these two boundary conditions: \(Y^{+}_{\tau ^{+}_{\eta }(\mu )} = Y^{-}_{\tau ^{-}_{\eta }(\mu )}\). This is the boundary-matrix comparison required for equality of the two restrictions crossing the periodic cut.