Dependencies

Embedding an operator on \(L\) consecutive sites into an \(N\)-site ring, where \(L\leq N\), does not increase its Hilbert-space operator norm.

For \(N\geq 2\), the cyclic embedding of a two-site operator at \(j\) belongs to \(\mathcal A_{e_j}\), including the bond crossing from the last site to the first.

Let \(H=\sum _j h_j\), with Hermitian \(h_j\in \mathcal A_{e_j}\) and \(\| h_j\| \leq 1\). Suppose \(A\in \mathcal A_X\) and \(B\in \mathcal A_Y\), where \(X,Y\) are nonempty and disjoint. For every \(a\geq 0\) and \(t\in \mathbb R\),

The constants are independent of \(N\) and \(d\). This is a nearest-neighbor finite-periodic specialization of Hastings–Koma, Appendix A, Theorem A.2; it is not the unrestricted interaction theorem.

Let \(h\) be a Hermitian two-site interaction with \(\| h\| \leq 1\), and let \(H_N=\sum _j h_{j,j+1}\) on the periodic chain of length \(N\geq 2\). For nonempty disjoint supports \(X,Y\) and \(A\in \mathcal A_X\), \(B\in \mathcal A_Y\), the estimate

holds for every \(a\geq 0\) and \(t\in \mathbb R\).

A permutation \(p\) of the sites sends an operator supported on \(X\) to an operator supported on \(p(X)\).

For fixed \(H,B,X\), the function \(t\mapsto C_B(X,t)\) is continuous.

For \(A\in \mathcal A_X\), one has \(\| [\tau _t(A),B]\| \leq C_B(X,t)\| A\| \).

The restricted commutator norm for \(H\) at time \(-t\) equals that for \(-H\) at time \(t\).

Let \(H=\sum _j h_j\) be a finite sum, where each \(h_j\) is Hermitian and belongs to \(\mathcal A_{T_j}\). For arbitrary \(B,X\) and \(t\geq 0\),

No bound on the ground energy or positivity assumption on \(h_j\) is required, and several terms may share a support. This is the recursion of Hastings–Koma, Appendix A, (A.12)–(A.14).

For arbitrary \(H,B,X\), one has \(C_B(X,0)\leq 2\| B\| \).

If \(B\in \mathcal A_Y\) and \(X\cap Y=\varnothing \), then \(C_B(X,0)=0\).

For nonempty \(X,Y\subseteq \Lambda \), let \(d(X,Y)\) be the least integer \(r\geq 0\) for which there exist \(x\in X\), \(y\in Y\), and \(m\in \mathbb Z\) such that \(|m|\leq r\) and \(y=x+m\pmod N\).

For nonempty supports and \(r\in \mathbb N\), no displacement of absolute value at most \(r\) connects \(X\) to \(Y\) if and only if \(r{\lt}d(X,Y)\).

Let \(f_i\geq 0\), for \(i\) in a finite set, be continuous on \([0,T]\) and satisfy \(f_i(t)\leq b_i+\sum _j a_{ij}\int _0^tf_j(s)\, ds\) with \(a_{ij}\geq 0\). If positive weights \(w_i\) satisfy \(\sum _ja_{ij}w_j\leq vw_i\) and \(b_i\leq cw_i\), then \(f_i(t)\leq cw_ie^{vt}\) for \(0\leq t\leq T\). If moreover \(b_i=0\) and \(v{\gt}0\), then \(f_i(t)\leq cw_i(e^{vt}-1)\).

If \(D {\gt} 0\), the factor system \(\omega \) from Theorem 10.1.21 satisfies the 2-cocycle identity for all \(g,h,k \in G\):

In the order \((m=0,+1,-1)\), the source tensor has letters

It is the diagonal physical rephasing \(\operatorname {diag}(1,1,-1)\) of Definition 15.3.7; this is a unitary basis change, not a virtual gauge. Take \(x=y=S_z=\operatorname {diag}(0,1,-1)\) and \(u=e^{i\pi S_z}=\operatorname {diag}(1,-1,-1)\). The rephasing leaves these physical operators unchanged, and the virtual gauge \(V=\sigma _z\) implements \(u\) on the printed letters.

Let \(S^x,S^y,S^z\) be the spin-\(1\) operators in the basis \(|0\rangle =|m=0\rangle \), \(|1\rangle =|m=+1\rangle \), \(|2\rangle =|m=-1\rangle \), with \(S^z=\operatorname{diag}(0,1,-1)\), \(S^+|0\rangle =\sqrt2|1\rangle \), \(S^+|2\rangle =\sqrt2|0\rangle \), \(S^x=\tfrac 12(S^++S^-)\) and \(S^y=\tfrac 1{2i}(S^+-S^-)\). This is the labelling in which the three matrices of the AKLT tensor of Definition 15.3.7 are proportional to \(\sigma ^z,\sqrt2\sigma ^+,-\sqrt2\sigma ^-\) on the indices \(0,1,2\). For two sites \(j,k\) of a chain let \(\vec S_j\cdot \vec S_k=\sum _\alpha S^\alpha _jS^\alpha _k\). On a periodic chain of \(N\) spin-\(1\) sites, with indices modulo \(N\), the AKLT Hamiltonian is

[ 2 , 3 ] ; see also [ 228 , local TeX lines 340–349 ] , and, up to an additive constant, [ 236 , lines 2126–2130 ] . Its two-site term is \(h=\vec S_0\cdot \vec S_1+\tfrac 13(\vec S_0\cdot \vec S_1)^2\).

Set \(d = 3\) (spin-1) and \(D = 2\). The AKLT tensor is the family \(\{ A^0, A^1, A^2\} \subset M_{2}(\mathbb {C})\) defined via the Pauli matrices by

Here \(\sigma _z = |0\rangle \! \langle 0| - |1\rangle \! \langle 1|\), so equivalently \(A^1 = \frac{1}{\sqrt{3}} \sigma _+\) and \(A^2 = -\frac{1}{\sqrt{3}} \sigma _-\) for \(\sigma _+ = \sqrt{2} |0\rangle \! \langle 1|\) and \(\sigma _- = \sqrt{2} |1\rangle \! \langle 0|\). Equivalently, the matrices are the Clebsch–Gordan coefficients coupling spin-\(1\) and spin-\(\tfrac 12\) to spin-\(\tfrac 12\). This is the representative of the tensor of Definition 15.3.1 used below (Lemma 15.3.8).

For a block of \(L\) physical sites, the induced action is the Kronecker-power representation

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

Set \(d = D = 2\). The cluster-state tensor of  [ 17 , Appendix A, “The cluster state” ] is the family \(\{ A^0, A^1\} \subset M_{2}(\mathbb {C})\) given by

Equivalently, \((A^s)_{\alpha \beta } = \delta _{\alpha s}\, (-1)^{s\beta }/\sqrt{2}\).

Take \(x=\sigma _z\otimes \sigma _y\) and \(y=\sigma _y\otimes \sigma _z\) on the two endpoint blocks. Blocked index \(i\) decodes to the ordered word \((i\bmod 2,\lfloor i/2\rfloor )\); the endpoint coefficient formulas use this order explicitly. On a two-site blocked chain the twist is \(u\otimes u\).

Two cocycles \(\omega _1,\omega _2\) are cohomologous (written \(\omega _1 \sim \omega _2\)) if there exists \(\varphi \colon G \to \mathbb {C}^{\times }\) such that, for all \(g,h \in G\),

The commutator phase of a \(2\)-cochain \(\omega \) at a pair \(g,h\) is \(\beta _\omega (g,h):=\omega (g,h) \omega (h,g)^{-1}\), defined for any \(\mathbb {C}^{\times }\)-valued \(2\)-cochain.

The local physical intertwining relation is \(\sum _j u_{ij}\, A^j = V\, A^i\, V^\dagger \). Locally, the physical action of \(u\) can therefore be pushed to the virtual legs:

= .

Transfer-map covariance is the identity that the transfer map commutes with virtual conjugation:

In doubled-layer notation, the virtual operators slide through the local transfer cell:

= .

In the transfer-map language, the commutation of the doubled transfer matrix \(E = \sum _i A_i \otimes \bar A_i\) with \(V \otimes \bar V\) is

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 \((i_1,\ldots ,i_L)\).

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, (8) 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 \(i\),

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 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 \(L\).

The second cohomology set is the quotient

where \(\sim \) is the coboundary-equivalence relation.

A tensor has physical string order if there exist a unitary \(u\notin \mathbb {C}\mathbb {1}\) and physical endpoint operators \(x,y\) for which Definition 10.3.1.6 holds. The nonscalar requirement is an explicit projective correction to [ 229 , display SOP, lines 114–121 ] , which prints only \(u\ne \mathbb {1}\). A scalar twist \(u=-\mathbb {1}\) with identity endpoints gives \(S_N=(-1)^N\) for every normalized unital tensor. The correction and the distinction from the literal source condition are recorded in [ 183 ] .

For fixed physical operators \(x,y,u\), the correlator has physical string order if there is \(s{\gt}0\) such that \(|S_N(x,y,u)|\to s\).

For boundary matrices \(X,Y\in M_{D}(\mathbb {C})\), set explicitly

The virtual-boundary nondecay predicate holds for \(u\) and \(\Lambda \) if there exist \(X,Y\) and a positive constant \(c{\gt}0\) such that \(c\le \| R_L(u;X,Y)\| \) for every \(L\). Thus some virtual-boundary sequence does not decay to zero.

This virtual-boundary condition asserts the existence of boundary matrices with non-vanishing transfer overlap under the twist \(u\). The relationship with physical endpoint operators (Definition 10.3.1.7) is analyzed in [ 183 ] .

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

An MPS tensor \(A\) is irreducible if it has no nontrivial invariant projection.

A scalar 2-cocycle \(\omega \) is a coboundary if there exists \(\varphi \colon G \to \mathbb {C}^{\times }\) such that, for all \(g,h \in G\),

A scalar 2-cochain \(\omega \colon G \times G \to \mathbb {C}^{\times }\) is a 2-cocycle if it satisfies, for all \(g,h,k \in G\),

For tensors \(A,B\) and a fixed physical representation \(U:G\to \mathrm{GL}_d(\mathbb {C})\), the predicate called same SPT phase asserts the existence of factor systems \(\omega _A,\omega _B:G\times G\to \mathbb {C}^\times \) and projective bond representations \(\rho _A,\rho _B\) such that

and \(\omega _A\sim \omega _B\).

The physical matrix of a tensor \(B=\{ B^i\} _{i=0}^{n-1}\) with \(B^i\in M_{D}(\mathbb {C})\) is the matrix \(\widehat B\in \mathbb {C}^{n\times D^2}\) with entries \(\widehat B_{i,(\alpha ,\beta )}=B^i_{\alpha \beta }\), the matrix of \(B\) as a map from the virtual pair space \(\mathbb {C}^{D^2}\) to the physical space \(\mathbb {C}^n\). Conversely, every \(R\in \mathbb {C}^{n\times D^2}\) is the physical matrix of the tensor \(B^i_{\alpha \beta }=R_{i,(\alpha ,\beta )}\).

The virtual local-covariance predicate holds for \(A,u,\Lambda \) if there exist a unitary matrix \(V\) and a phase \(\mu \) with \(|\mu | = 1\) such that \(V^\dagger \Lambda V = \Lambda \) and, for every physical index \(i\),

The unitary \(V\) intertwines the physical action of \(u\) with a virtual conjugation, and the boundary state \(\Lambda \) is invariant under \(V\).

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 MPV inner product between two tensors \(A\) and \(B\) at system size \(N\) is

i.e. the Euclidean inner product on \(\mathbb {C}^{d^N}\) (conjugate-linear in the first argument).

The MPV overlap between two tensors \(A\) (of bond dimension \(D_1\)) and \(B\) (of bond dimension \(D_2\)) at system size \(N\) is

Diagrammatically, the overlap contracts the MPV ring of \(A\) against the conjugate ring of \(B\) along their shared physical indices, both virtual rings closed by the trace:

This pairing is linear in the first MPV coefficient and conjugate-linear in the second. By Lemma 2.8.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.8.2.

We regard \(|V^{(N)}(A)\rangle \) as an element of \(\mathbb {C}^{d^N}\) indexed by configurations \(\sigma \in \{ 0,\ldots ,d{-}1\} ^N\), with \(\sigma \)-component \(V^{(N)}(A)_\sigma \) as in Definition 2.1.7. This allows us to use the standard Euclidean inner product on \(\mathbb {C}^{d^N}\).

A factor system \(\omega \) has a non-trivial class when it is not cohomologous to the constant cocycle \(1\).

A tensor \(A\) is normal [ 17 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of  [ 13 ] , (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 \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

An MPS tensor \(A\) has on-site symmetry under a representation \(U:G\to \mathrm{GL}_d(\mathbb {C})\) if \(\mathcal{V}(A)=\mathcal{V}(\widetilde A_g)\) for every \(g\in G\). This equality of periodic MPV families provides the condition for constructing virtual symmetry gauges.

Let \(U\) be a linear representation of a group \(G\) on the physical space and let \(\phi :G\to \mathbb {C}^\times \) be a character. A tensor \(A\) is on-site symmetric up to \(\phi \) if, for every \(g\in G\), its twist \(\widetilde A_g\) is gauge equivalent to \(\phi (g)A\). In particular,

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

The parent interaction at range \(L\) is the canonical orthogonal projector specified in (14).

For an observable \(O\) on a block of \(L\) physical spins, define the linear map on the virtual matrix space by

This transfer map also depends linearly on the observable \(O\). Diagrammatically, \(O\) is inserted between the two layers of the doubled chain on its \(L\) sites:

This is the observable transfer map of [ 13 , lines 490–496 ] .

For a matrix \(\Lambda \in M_{d}(\mathbb {C})\) and a tensor \(A=(A^i)_{i=0}^{d-1}\), the physically rotated tensor \(\Lambda A\) has matrices

For physical operators \(x,y \in M_{d}(\mathbb {C})\), a twist \(u \in M_{d}(\mathbb {C})\), and boundary state \(\Lambda \), the transfer-form string correlator of [ 229 , display SOPMP, lines 176–181 ] is

This is the physical-endpoint expression corresponding to \(\langle \Psi |\, x \otimes u^{\otimes N} \otimes y\, |\Psi \rangle \).

Given a scalar 2-cochain \(\omega \), a projective representation of \(G\) on \(\mathbb {C}^D\) is a map \(\rho : G \to \mathrm{GL}_{D}(\mathbb {C})\) satisfying, for all \(g,h \in G\),

Two projective representations are projectively equivalent when their factor systems are cohomologous in \(H^2(G,\mathbb {C}^\times )\).

Given a matrix \(M \in \mathbb {C}^{m\times d}\) and an MPS tensor \(A\) with physical dimension \(d\), the physical-index rotation is the tensor with physical dimension \(m\) given by \((A_M)^i := \sum _j M_{ij}\, A^j\) for \(i=0,\ldots ,m-1\). When the matrix is a unitary \(u\), the rotation is written \(A_u\).

For \(A=\{ A^i\} _{i=0}^{d-1}\), let \(F:\mathbb {C}^d\to \mathbb {C}^{D\times D}\) have column \(i\) equal to the row-vectorization of \(A^i\):

The row-vectorization convention is \((\operatorname{vec}_r X)_{(a,b)}=X_{ab}\). Thus \(E\operatorname{vec}_r X=\operatorname{vec}_r\mathcal{E}_A(X)\).

Two tensors \(A\) and \(B\) (of the same bond dimension) generate the same MPV family if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every system size \(N\).

A scalar 2-cochain on a group \(G\) is a function \(\omega : G \times G \to \mathbb {C}^{\times }\).

For the periodic length-\(L\) matrix product vector \(|\Psi _L(A)\rangle \), the source quantity is literally

For the raw periodic matrix product vector, this overlap is \(\operatorname {Tr}(\mathcal E_u^L)\). Dividing by its squared norm gives \(\operatorname {Tr}(\mathcal E_u^L)/\operatorname {Tr}(\mathcal E^L)\) at nonzero lengths. The stationary-boundary functional is instead a fixed-support thermodynamic limit with an untwisted exterior; the full-ring support grows with the ambient ring length. The numerator has two separately closed virtual rings:

The two return paths differ from the stationary \(\Lambda \)-cup closure in Definition 10.3.2. The identity insertion gives the same periodic closure for \(Z_L\).

Let \(\mathcal H_D=M_{D}(\mathbb {C})\) have the Hilbert–Schmidt inner product, and let \(e_D:\mathcal H_D\to \mathbb {C}^{D^2}\) be column vectorization. For \(V\in M_{D}(\mathbb {C})\), define the linear operator \(\beta _V=e_D\circ (X\mapsto VXV^\dagger )\circ e_D^{-1}\).

Let \(U\) be a monoid representation by unitary matrices on \(\mathbb {C}^d\). For \(L\geq 0\), its unitary action on the blocked physical space is \(U^{[L]}(g)=U(g)^{\otimes L}\), in the coordinates of length-\(L\) words.

Define \(\Psi _C=\Phi _C^\dagger (\Phi _C\Phi _C^\dagger )^{-1}\). Here the inverse is assigned the value zero when the operator is not invertible.

For a tensor \(C=\{ C^i\} _{i=0}^{d-1}\) with bond dimension \(D\), let \(\Phi _C:\mathbb {C}^d\to \mathbb {C}^{D^2}\) be the operator with matrix entries \((\Phi _C)_{(a,b),i}=C^i_{b,a}\).

For projective representations \(\rho _0:G\to \mathrm{GL}(D_0,\mathbb {C})\) and \(\rho _1:G\to \mathrm{GL}(D_1,\mathbb {C})\), set \(S_g=\overline{\rho _0(g)}\otimes \rho _1(g)\) on \(\mathbb {C}^{D_0}\otimes \mathbb {C}^{D_1}\). Under column vectorization this is \(X\mapsto \rho _1(g)X\rho _0(g)^\dagger \) on rectangular matrices.

Put \(K=D_0+D_1\) and let \(J_j:\mathbb {C}^{D_j^2}\to \mathbb {C}^{K^2}\) be the two coordinate inclusions. On \(H=\mathbb {C}^{K^2}\oplus \mathbb {C}^{d_0}\oplus \mathbb {C}^{d_1}\), take the action \((V\oplus U_0)\oplus U_1\), where \(V\) is the fixed-point pair action of the virtual direct sum, with trivial scalar character. Write \(P_{m,d}:\mathbb {C}^m\to \mathbb {C}^m\oplus \mathbb {C}^d\) for the first-summand inclusion; thus \(P_{m,d}W=(W,0)\). If \(F_j\) is the isometric polar factor of a one-site injective tensor \(A_j\), define

These retain the original physical complements in the common-space construction of [ 236 , Section II.F.2 ] . The injective and character restrictions are recorded in [ 176 ] and [ 186 ] .

For two one-site injective tensors of positive bond dimension with unitary physical and virtual actions giving exact adjoint covariance and the same virtual factor system, the constructed common-space path has a continuous exact MPS ground realization. It joins the two included endpoint vector lines through the polar deformations and the canonical fixed-point comparisons.

The two common-space polar parent paths have exact MPS ground realizations in the common bond space of dimension \(D_0+D_1\). For endpoint \(j\), include the scaled polar tensor family by the corresponding coordinate isometry and take its supported lift with initial tensor the included weighted fixed-point endpoint.

Under the hypotheses of the common-embedding theorem, each canonical two-site parent of \(E_jA_j\) is joined to the canonical parent of the normalized matrix-unit tensor in the frame \(QJ_j\) by a symmetric gapped path under the same common physical action. The gap is positive uniformly in the parameter and every ring length \(N\ge 2\); the local interaction has norm at most one.

Fix a unitary representation \(U:G\to \mathrm U(d)\) and two two-site interactions \(h_0,h_1\). A symmetric gapped path between them on this common physical space consists of Hermitian interactions \(h_\gamma \), \(0\le \gamma \le 1\), with the prescribed endpoints, such that:

1. \(\| h_\gamma \| _{\mathrm{op}}\le 1\);

2. \(\gamma \mapsto h_\gamma \) is continuous;

3. for every periodic chain of length \(N\ge 2\), the Hamiltonian \(H_N(\gamma )=\sum _i h_\gamma (i,i+1)\) commutes with \(U_g^{\otimes N}\);

4. there is a \(\delta {\gt}0\), independent of \(N\) and \(\gamma \), such that every spectral value of \(H_N(\gamma )\) above its least spectral value \(E_N(\gamma )\) is at least \(E_N(\gamma )+\delta \).

Ground-state degeneracy is allowed. This is the common-space path condition of  [ 236 , Sections II.C.1–2, lines 407–453 ] ; the choice of endpoint blocking and symmetry-compatible embeddings precedes this definition. No condition on virtual cocycles enters it.

An injection \(e:\{ 1,\ldots ,D\} \to \{ 1,\ldots ,K\} \) defines an isometry \(J_e\) by \(J_e|a\rangle =|e(a)\rangle \). Its adjoint selects the included coordinates, so \(J_e^\dagger M J_e\) is the corresponding submatrix. If the rows of \(M\) are supported on these coordinates, then \(J_eJ_e^\dagger M=M\) and \(MJ_e=J_e(J_e^\dagger M J_e)\). The physical inclusion \(E_e\) sends \(|a,b\rangle \) to \(|e(a),e(b)\rangle \). It has orthonormal columns, and the embedded fixed-point tensor has entries

For \(D{\gt}0\) this tensor is injective [ 236 , Section II.F.2 ] .

Suppose \(D_0,D_1{\gt}0\) and the virtual representations are unitary with the same factor system. Use the common physical action with trivial character. The canonical two-site parents of \(B_0\) and \(B_1\) are connected on \(\mathbb {C}^{D_0+D_1}\otimes \mathbb {C}^{D_0+D_1}\) by a continuous symmetric path of interactions of norm at most one. Their periodic Hamiltonians have a positive uniform gap for all chain lengths \(N\geq 2\) [ 236 , Section II.F.2 ] .

Let \(D_0,D_1{\gt}0\) and \(K=D_0+D_1\). Let \(e_0,e_1\) be the two coordinate inclusions into \(\mathbb {C}^K\), and let \(A_t\) be the weighted matrix-unit tensor. Define

These are the physical embeddings \(E_{e_0}A^{(D_0)}\) and \(E_{e_1}A^{(D_1)}\) of the standard fixed points; their bond dimensions remain \(D_0,D_1\). Both are injective, \(G_L(B_j)\subseteq G_L(A_j)\), and their periodic vectors equal those of \(A_j\) at every positive length. For projective representations \(\rho _0,\rho _1\) with the same factor system, the common physical action of \(\rho _0\oplus \rho _1\) with trivial character acts on \(B_j\) by conjugation with \(\rho _j(g^{-1})\) [ 236 , Section II.F.2 ] .

Let \(A_\gamma \) be a continuous family of one-site injective tensors with bond dimension \(D{\gt}0\), and let \(U\) and \(V\) be unitary on-site representations on \(\mathbb {C}^d\) and \(\mathbb {C}^m\). Suppose \(E^\dagger E=I\), \(V_gE=EU_g\) for every \(g\), and each \(U_gA_\gamma \) is a virtual gauge change of \(A_\gamma \). Applying the canonical injective-path construction to \(EA_\gamma \) gives a symmetric gapped interaction path from \(h_2(EA_0)\) to \(h_2(EA_1)\) under the prescribed action \(V\). In particular, when the covariance of \(A\) is implemented by unitary bond matrices, its polar deformation gives a path from \(h_2(EW)\) to \(h_2(EA)\), where \(W\) is the polar isometric form of \(A\). These are the physically embedded endpoint deformations used in [ 236 , Sections II.C and II.F.2 ] . The one-site injective restriction is documented in [ 176 ] .

Two consecutive physical interaction paths with exact MPS ground realizations can be concatenated when their ambient bond dimensions agree and their endpoint tensors coincide under that identification. The resulting continuous tensor path realizes the ground lines of the concatenated physical interaction path.

Three exact MPS ground realizations with a common ambient bond dimension can be joined after reversing the first path, provided their tensors agree at the two joins. This gives the continuous ground realization of the same three-stage physical path.

Two tensors satisfy the continuous exact MPS phase condition if, after a common positive blocking length, their whole physical spaces admit isometric inclusions as orthogonal summands of one common physical representation, as required in [ 236 , Section II.C.2 ] . Multiplication of each blocked endpoint action by a unit-modulus character is allowed. The inclusions intertwine these actions with the common action, and the included canonical two-site parents are joined by a symmetric gapped interaction path.

Along that path a continuous tensor family of some fixed finite ambient bond dimension represents the unique least-energy line at every ring length \(N\ge 2\). Its periodic vectors are nonzero at every positive length. At each parameter its positive-length vector lines have a one-site injective representative of positive bond dimension bounded by the ambient dimension; this dimension may vary. The endpoint vector lines agree with those of the included tensors.

This describes the continuous exact MPS regime of [ 236 , Section II.F.2 ] . It assumes no virtual action, factor system, or continuous canonical bond data. The classification implication, including changes of minimal bond dimension, remains open as recorded in [ 101 ] .

Let \(\rho \) be a projective representation of \(G\) on \(\mathbb {C}^D\) with factor system \(\omega \), let \(\phi \) be a character of \(G\), and put \(W_g=\rho (g^{-1})\). The SPT fixed-point tensor has physical space \(\mathbb {C}^D\otimes \mathbb {C}^D\) and letters \(A^{(a,b)}=D^{-1/2}|a\rangle \langle b|\), and \(G\) acts on the physical space by \(U(g)=\phi (g)\, W_g^{T}\otimes W_g^{-1}\). The projective representation need not be unitary; when it is, \(W_g^{-1}=W_g^{\dagger }\), so with \(X_g=W_g^{\dagger }\) the conjugation \(W_gA^iW_g^{-1}\) is the source’s \(X_g^{\dagger }A^iX_g\).

Let \(F:S\to H\) and \(J:S\to K\) be linear isometries between finite-dimensional complex Hilbert spaces. Define \(E:H\to K\oplus H\) by

For an injective tensor \(A\), choose \(F\) to be its isometric polar factor. The first summand then carries the prescribed target frame, while the second retains the original orthogonal complement. This is an auxiliary construction for the common physical representation in [ 236 , Section II.F.2 ] .

Let each site of a cyclic chain carry two registers of dimension \(D\). For a configuration \(s=((L_i,R_i))_i\), define

The inverse recovers \((L_i,R_i)=((b_i)_2,(b_{i+1})_1)\). Pair coordinates may equivalently be enumerated by an alphabet of size \(D^2\). These are the independent-bond coordinates of  [ 236 , Section II.D.2 ] .

Let \(E:H\to H'\) be a one-site isometry and put \(T=E\otimes E\). For a two-site interaction \(h\), define

This gives energy one on the orthogonal complement of the included two-site space, as in the physical embedding construction of [ 236 , Section II.F.2 ] .

For a bilinear product \(\mu \), a frame \(J\) of a principal left ideal, and a left inverse \(H\), define \(L(x)v=H\mu (x,Jv)\).

The coefficients are the unit circle \(\mathrm U(1)\), with action \(C_p\). A twisted character and factor system satisfy

A unitary conjugate-projective representation has

Associativity forces the displayed two-cocycle equation. Rephasing \(X_g\) by \(\xi (g)\in \mathrm U(1)\) changes the factor system by

These are Mathlib’s multiplicative cocycles and coboundaries for the specified coefficient action. Their classes belong to \(H^1_t(G,\mathrm U(1))\) and \(H^2_{t+r}(G,\mathrm U(1))\).

Define \(T_{t,r}(M)=C_t(M)\) for \(r=0\), and \(T_{t,r}(M)=C_t(M)^{\mathsf T}\) for \(r=1\). The mixed physical action is

A tensor has mixed symmetry with phase \(\phi (g)\) and virtual gauge \(X_g\) when \(S_g(A)^i=\phi (g)X_g^\dagger A^iX_g\).

Let \(a,b\) be positive two-site interactions on one physical space, of operator norm at most one, with \(a\leq b\). Suppose that for every periodic length \(N\geq 2\) their Hamiltonians \(H_N(a)\) and \(H_N(b)\) have the same nonzero ground space at zero energy, and both commute with the same on-site symmetry. If \(H_N(a)\) has a spectral gap at least \(\delta {\gt}0\) for all \(N\geq 2\), then

is a symmetric gapped interaction path from \(a\) to \(b\), with gap at least \(\delta \). This comparison applies to independent-bond and canonical parent interactions in the fixed-point construction [ 236 , Section II.F.2 ] .

The space \(\mathbb {C}^{K^2}\oplus \mathbb {C}^{d_0}\oplus \mathbb {C}^{d_1}\) has three sectors with left dimensions \((K,d_0,d_1)\) and right dimensions \((K,1,1)\). Within the first sector, the pair \((r,l)\) is enumerated in physical order \((l,r)\). The other two sectors occupy the middle and final summands. These are auxiliary coordinates for the enlarged physical space in [ 236 , Section II.F.2 ] .

Let \(M:\mathbb {C}^{D^2}\to \mathbb {C}^d\) be the physical map of a tensor \(A\), and write \(M=VP\) for its right polar decomposition. Define \(W\) by its physical map \(V\), and define \(A_\gamma \) by

This is the polar interpolation of [ 236 , Section II.C ] , expressed in right polar factors. It stays in the original physical space; the map \(M\) may be rectangular.

Let \(A\) be one-site injective with bond dimension \(D{\gt}0\) and unitary covariance \((U_g A)^i=X_gA^iX_g^\dagger \). Let \(J:\mathbb {C}^{D^2}\to K\) be an isometry satisfying \(V_gJ=J(X_g^T\otimes X_g^\dagger )\) for a unitary on-site representation \(V\). Choose the embedding \(E\) in (67) using the polar isometric factor \(F\) of \(A\). Embedding the polar deformation gives a symmetric gapped path under \(V\oplus U\) from the canonical two-site parent of the tensor with frame \((J,0)\) to the canonical two-site parent of \(EA\). The gap is positive uniformly in the path parameter and every ring length \(N\ge 2\). This is the prescribed-frame version of the embedded polar construction in [ 236 , Sections II.C and II.F.2 ] . The one-site injective restriction is documented in [ 176 ] .

Let \(A\) be one-site injective with \(D{\gt}0\), let \(U\) be a unitary on-site representation, and suppose the tensor covariance is expressed by unitary bond conjugation. The canonical two-site parent interactions of \(A\) and of its isometric form \(W\) are joined by a symmetric gapped interaction path on the original physical space, with the same representation \(U\) throughout [ 236 , Section II.C, isometric form and symmetries ] . No equality between the physical dimension and \(D^2\) is required.

Let \(V^0\) and \(V^1\) be projective representations on \(\mathbb {C}^{D_0}\) and \(\mathbb {C}^{D_1}\) with the same factor system \(\omega \). Their direct sum is the projective representation on \(\mathbb {C}^{D_0+D_1}\) given by

This is the virtual direct sum in the symmetric interpolation of  [ 236 , Section II.F.2, lines 909–921 ] .

Set \(R_D^0=I_D\) and \(R_D^1=0\).

In a decomposition \(H=\bigoplus _q(H_{q,r}\otimes H_{q,l})\), prescribe a neighboring matrix \(\eta _{q,h}\) on \(H_{q,r}\otimes H_{h,l}\). The two-site interaction is the direct sum of \(I_{q,l}\otimes \eta _{q,h}\otimes I_{h,r}\), expressed in the original physical coordinates. For the three-sector common physical space, choose \(\eta _{0,0}=B\) and \(\eta _{q,h}=I\) for every other pair. The active pair then carries the independent virtual bond, while a neighboring pair meeting either additional physical sector carries the identity penalty. These are auxiliary interactions for [ 236 , Sections II.D.2 and II.F.2 ] .

Let \(V:\mathbb {C}^D\to \mathbb {C}^E\) be an isometry, let \(A_\gamma \) be a tensor family, and let \(T\) be a tensor on \(\mathbb {C}^E\). The isometric bond inclusion of \(A\) has letters \(VA^iV^\dagger \). The supported lift with prescribed initial tensor \(T\) has letters

For an isometry \(K:\mathbb {C}^D\to \mathbb {C}^k\) and a matrix \(X\in M_{D}(\mathbb {C})\), set \(\widetilde X=KXK^\dagger +(I-KK^\dagger )\).

For the two-site trace form \(G\) and an injective section \(F\), set \(q_i=((GF)^\dagger GF)^{-1}(GF)^\dagger G e_i\).

Let \(G\in M_d(\mathbb {C})\) be the complex-bilinear two-site trace form, let \(F:\mathbb {C}^r\to \mathbb {C}^d\) be a coefficient section, and put \(C=GF\). For contracted three-site columns \(R:\mathbb {C}^{r\times r}\to \mathbb {C}^d\), set

The positive Hermitian matrix \(C^\dagger C\) serves to invert the section. No positivity of the bilinear trace form \(G\) is assumed.

For an ambient tensor \(B\) and a coefficient section \(F\), define

The coefficient matrix across the one-site cut of the periodic two-site MPS is \(C_2(A)_{ij}=\operatorname {tr}(A^iA^j)\).

Suppose the two virtual summands carry unitary projective representations with the same factor system. Let \(h_t\) be the bond interaction and \(p_t\) the canonical parent projection of the weighted tensor. For each fixed real \(t\), the affine path \(s\mapsto h_t+s(p_t-h_t)\) is a symmetric gapped path from \(h_t\) to \(p_t\). Concatenating its reverse at \(t=0\), the bond path from \(h_0\) to \(h_1\), and the comparison at \(t=1\) gives a symmetric gapped path from \(p_0\) to \(p_1\) on the common direct-sum physical space [ 236 , Section II.F.2 ] .

Let \(A\) be a tensor of bond dimension \(D\) and let \(\Lambda \in M_{D}(\mathbb {C})\). For words \(s,t\) of length \(N\), define the physical matrix

The upper and lower physical legs remain open, while the virtual boundary is contracted with \(\Lambda \):

.

For a boundary state \(\Lambda \) and boundary matrices \(X,Y \in M_{D}(\mathbb {C})\), the virtual-boundary twist functional is

The special case \(X=Y=\mathbb {1}\) is the stationary-boundary block-twist functional of Definition 10.3.2. The matrices \(X,Y\) act on the virtual level. In the physical string correlator of  [ 229 ] the string of twists is instead flanked by local operators \(x,y\) acting on physical sites; expanding such operators in the transfer picture produces particular virtual matrices, and [ 229 ] argues that for injective tensors, physical operators on sufficiently many sites produce every virtual matrix. The results below are stated directly in this virtual-boundary form.

The stationary-boundary block-twist functional is

In tensor-network notation this is the length-\(L\) doubled chain with a copy of \(u\) inserted on each physical rung:

.

The boundary matrix \(\Lambda \) is contracted into the virtual boundary, the red nodes labelled \(u\) sit on the physical rungs, and the right boundary is traced.

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.

For an MPS tensor \(A\) and a physical-index matrix \(u\), define the companion tensor \(B_u\) by

Given a group representation \(U : G \to \mathrm{GL}_{d}(\mathbb {C})\) on the physical index, the \(g\)-twisted tensor is defined, for each \(i \in \{ 0,\ldots ,d{-}1\} \), by

Given an MPS tensor \(A\) and a physical-index matrix \(u\), the twisted transfer map is

Diagrammatically one inserts the physical operator \(u\) on the doubled local transfer cell:

.

The upper and lower black nodes denote \(A_n\) and \(A_{n'}^\dagger \), and the red node labelled \(u\) couples the physical legs.

If \(\rho (\mathcal{E}_u) {\lt} 1\), then, as \(L \to \infty \), for all boundary matrices \(X,Y,\Lambda \in M_{D}(\mathbb {C})\) one has \(R_L(u;X,Y) \longrightarrow 0\).

Suppose \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), \(\Lambda {\gt}0\), and \(\mathcal{E}_A^*(\Lambda )=\Lambda \). If every fixed matrix of \(\mathcal{E}_A\) is a scalar multiple of \(\mathbb {1}\), then \(\mathcal{E}_A\) is irreducible.

For every virtual matrix \(X\),

Apply the existing FNW complementary-remainder argument to the conjugate-transposed Kraus family. No diagonalizability or explicit eigenbasis of the transfer map is assumed.

If a matrix \(Z \in M_{D}(\mathbb {C})\) commutes with every element of \(M_{D}(\mathbb {C})\), then there exists \(\lambda \in \mathbb {C}\) such that \(Z = \lambda \mathbb {1}_D\).

If \(\rho \) is a projective representation with factor system \(\omega \) and \(D {\gt} 0\), then \(\omega \) satisfies the 2-cocycle identity

If \(B^i = X A^i X^{-1}\) for all \(i\), then \(B^w = X A^w X^{-1}\) for every word \(w\).

Let \(f,g\) be invertible operators on a finite-dimensional complex vector space \(V\) with \(fg=-gf\). Then \(\dim V\) is even. Consequently, if a matrix \(\Lambda \) commutes with two anticommuting invertible matrices \(X\) and \(Y\), every eigenspace of \(\Lambda \) has even dimension.

If \(A\) is injective and on-site symmetric under \(U\), then for every \(g \in G\) the twisted tensor \(\widetilde{A}_g\) is gauge equivalent to \(A\).

Let \(A\) be an injective tensor with \(D\ge 1\). If \(X,Y\in \mathrm{GL}_{D}(\mathbb {C})\) and \(c\in \mathbb {C}\) satisfy \(X^{-1}A^iX=c\, Y^{-1}A^iY\) for all \(i\), then \(c=1\) and \(X=u\, Y\) for some \(u\in \mathbb {C}^{\times }\) [ 17 , Section III.A ] .

Let \(A\) be a normal tensor with \(D\ge 1\). If \(X,Y\in \mathrm{GL}_{D}(\mathbb {C})\) and \(c\in \mathbb {C}\) satisfy \(X^{-1}A^iX=c\, Y^{-1}A^iY\) for all \(i\), then \(c=1\) and \(X=u\, Y\) for some \(u\in \mathbb {C}^{\times }\) [ 17 , Section III.A ] .

If \(X,Y\in \mathrm{GL}_{D}(\mathbb {C})\) and \(c\in \mathbb {C}\) satisfy \(X^{-1}A^iX=c\, Y^{-1}A^iY\) for all \(i\), then for every word \(w\),

Let \(A\) be an injective tensor. If \(X,Y \in \mathrm{GL}_{D}(\mathbb {C})\) both satisfy \(B^i=X\, A^i\, X^{-1}=Y\, A^i\, Y^{-1}\) for all \(i\), then there exists a nonzero scalar \(u \in \mathbb {C}^{\times }\) such that \(Y=u\cdot X\).

Assume \(\operatorname{tr}(\Lambda )=1\) and \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\). If \(A,u,\Lambda \) satisfy virtual local covariance, then virtual-boundary nondecay holds for \(u\).

Let \(A\) be normal with \(D\ge 1\) and let \(X\) be unitary. If \(A^i=c\, (X\bar X)^\dagger A^i(X\bar X)\) for all \(i\), then \(X\bar X=\pm \mathbb {1}\). If two unitaries \(X,X'\) satisfy \(\zeta X^\dagger A^iX=\zeta 'X'^\dagger A^iX'\) for all \(i\) with \(\zeta \neq 0\), then \(X\bar X=X'\bar X'\).

Let \(A\) be unital and irreducible, and let \(u\) be a physical unitary. A common positive gauge puts \(A\) and its unitary-mixed companion in trace-preserving form. Every eigenvalue of \(\mathcal{E}_u\) has modulus at most \(1\). A modulus-one eigenvalue gives gauge-phase equivalence; the gauge can be normalized to a unitary \(V\). For a faithful normalized stationary dual density, \(V^\dagger \Lambda V=\Lambda \). The earlier injective interfaces are specializations of these results.

A tensor \(A\) is \(L\)-block injective if and only if its \(L\)-blocked tensor \(A^{[L]}\) is injective.

For operators \(X,X'\) on \((\mathbb {C}^d)^{\otimes L_1}\) and \(Y,Y'\) on \((\mathbb {C}^d)^{\otimes L_2}\), with \(X\otimes Y\) acting on \(L_1+L_2\) consecutive sites,

For an operator \(O\) on \((\mathbb {C}^d)^{\otimes N}\) and the unnormalized vector \(|\phi _N(A)\rangle \) of (1), \(\langle \phi _N(A)|O|\phi _N(A)\rangle =\operatorname {tr}\mathcal{E}_O\), the trace of a linear map on \(M_{D}(\mathbb {C})\).

For \(W\in \mathbb {C}^{m\times n}\), the physical matrix of \(B_W\) is \(W\widehat B\).

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for every \(N\) and every configuration \(\sigma = (i_1, \ldots , i_N)\), \(V^{(N)}(\zeta A)_\sigma = \zeta ^{N} V^{(N)}(A)_\sigma \).

If \(A\) has on-site symmetry under \(U\), then so does \(cA\) for every \(c\in \mathbb {C}\), and so does every tensor gauge equivalent to \(A\).

For \(\Lambda ,\Lambda '\in M_{d}(\mathbb {C})\) one has \((\Lambda \Lambda ')^{\otimes L}=\Lambda ^{\otimes L}\Lambda '^{\otimes L}\), \(\mathbb {1}^{\otimes L}=\mathbb {1}\), and \((\Lambda ^{\otimes L})^\dagger =(\Lambda ^\dagger )^{\otimes L}\). In particular, \(\Lambda ^{\otimes L}\) is unitary when \(\Lambda \) is. If \(\Lambda \) is invertible, then \((\Lambda ^{-1})^{\otimes L}\) is a two-sided inverse of \(\Lambda ^{\otimes L}\).

\(O_{AB}(N) = \overline{\langle V^{(N)}(A) | V^{(N)}(B) \rangle }\).

Let \(B\) and \(C\) satisfy \(\sum _iB_iB_i^\dagger =\mathbb {1}\) and \(\sum _iC_iC_i^\dagger =\mathbb {1}\), let \(B\) be \(L_0\)-block injective, and let \(R\neq 0\) and \(x\neq 0\) satisfy (26). Then \(R\) is invertible, \(|x|=1\), and there is a unitary \(U\) with

Consequently the length-\(N\) coefficients of \(B\) and \(C\) satisfy \(V^{(N)}(B)=x^NV^{(N)}(C)\) for every \(N\).

This is Lemma 1 of  [ 229 ] . Let \(A\) represent a pure finitely correlated state in the canonical gauge \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), \(\mathcal{E}_A^*(\Lambda )=\Lambda \), with \(\Lambda {\gt}0\) and \(\operatorname{tr}(\Lambda )=1\). For every physical unitary \(u\),

Equality holds if and only if there are a unitary \(V\) and \(\theta \in [0,2\pi )\) satisfying the following identity for every \(j\) in an eigenbasis \(u=\sum _j e^{i\theta _j}|\widetilde j\rangle \langle \widetilde j|\):

Moreover, \(\mathcal{E}_u\) has at most one eigenvalue of modulus one.

Let \(W\) contain physical eigenbasis bras, so \(Wu=\operatorname {diag}(\zeta _j)W\), and set \(\widetilde A_j=\sum _k W_{jk}A_k\). The basis-independent covariance equation is equivalent to

Taking \(\mu =e^{i\theta }\) and \(\zeta _j=e^{i\theta _j}\) gives exactly \(e^{i(\theta -\theta _j)}\) in the source. The equality criterion also supplies the source representative \(0\leq \theta {\lt}2\pi \). The phases determine the identity independently of other representatives modulo \(2\pi \).

For \(\Lambda ,\Lambda '\in M_{d}(\mathbb {C})\),

For unitary \(u\), the shifted twist \(u_0=\mu ^{-1}u\) is unitary and has the unphased virtual intertwiner \(V\). Taking \(y=x^\dagger u_0\) gives

The proof uses the actual physical matrix product in the endpoint: \(\mathcal{E}_{x^\dagger u_0}(\mathbb {1})=V\mathcal{E}_{x^\dagger }(V^\dagger )\). It requires only Hermiticity of \(\Lambda \) for the trace conjugation.

Scalar rephasing obeys

Hence \(|\mu |=1\) preserves the fixed-endpoint positive-limit condition. If \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\) and \(\operatorname{tr}\Lambda =1\), then \(S_N(\mathbb {1},\mathbb {1},\mu \mathbb {1})=\mu ^N\). Thus \(u=-\mathbb {1}\) satisfies the literal nonidentity-twist condition whenever \(d{\gt}0\). Under the simple-peripheral purity property, a scalar unitary twist possessing a nonzero fixed twisted eigenmatrix must instead be the identity.

For any \(A,\Lambda ,x,y,u\), if \(\rho (\mathcal{E}_u){\lt}1\), then \(S_N(x,y,u)\to 0\). Expand the left endpoint insertion into its finite matrix sum and apply virtual-boundary decay to each term after trace cyclicity.

Let \(\rho \) be a projective representation of \(G\) on \(\mathbb {C}^D\) whose factor system \(\omega \) is not cohomologous to the trivial one, and let \(\Lambda \) commute with every \(\rho (g)\). Then no eigenspace of \(\Lambda \) is one-dimensional.

Let \(A\) be injective and let invertible matrices \(X_g\) implement the physical twists, \(U_g(A)=X_g A X_g^{-1}\). Then \(\rho (g)=X_{g^{-1}}\) is a projective representation for some scalar factor system \(\omega \). Thus chosen unitary matrices give a unitary projective representation [ 236 , Section II.F.2 ] .

If a normalized normal tensor has bond dimension \(D\le M\), its \(M^4\)-site blocked tensor is injective.

For a normalized normal tensor of bond dimension \(D\), products of length \(D^4\) span the full matrix algebra. Source context: arXiv:1606.00608, lines 318–344, and the quantum Wielandt bound.

If two tensors have equal periodic rays at every positive length, then their \(L\)-site blocked tensors have equal periodic rays at every positive length, for every \(L{\gt}0\).

Let \(T\) be a positive unital linear map on \(M_D(\mathbb C)\), with \(D{\gt}0\). If its fixed space is one-dimensional, the eigenvalue \(1\) has algebraic multiplicity one. This is the fixed-eigenvalue consequence of Wolf, Proposition 6.2.

Let \(\eta \) be a unit bond vector and \(N\geq 1\). The local penalty \(1-|\eta \rangle \langle \eta |\) has operator norm at most one, and the independent-bond Hamiltonian \(H_N(\eta )\) is positive semidefinite with

For \(D_0,D_1{\gt}0\) and the normalized interpolating bond \(\widehat\omega (\gamma )\), the penalty depends continuously on \(\gamma \), and these statements hold for every real \(\gamma \) and every \(N\geq 1\), including both endpoints [ 236 , Section II.F.2, lines 899–906 ] .

The letter actions compose by addition of both parities, but \(T_{t,r}(zM)=C_t(z)T_{t,r}(M)\). For every matrix \(X\),

Consequently physical matrices compose as \(U_gC_{t(g)}(U_h)\), whereas virtual gauges compose as \(X_gC_{t(g)+r(g)}(X_h)\). Reflection reverses the order in a product of letters. Thus on a chain the action is the on-site matrix action together with complex conjugation when \(t(g)=1\) and reversal of the physical site order when \(r(g)=1\).

Let \(P_N\) be the \(d^N\times D^2\) matrix with entries \(P_N(s,(a,b))=(A_s)_{ab}\), and let \(K=\Lambda ^{\mathsf T}\otimes \mathbb {1}_D\). Then \(\varrho _N(A,\Lambda )=P_NKP_N^\dagger \).

For an MPS tensor \(A\) and bond matrices \(\sigma , \rho \in M_{D}(\mathbb {C})\),

where \(\mathcal{E}_{A^\dagger }\) is the transfer map with Kraus operators \((A^i)^\dagger \).

For every \(g\in G\) and \(L\ge 0\),

If \(R\in \mathrm{GL}_{D}(\mathbb {C})\) satisfies \((\widetilde A_g)^i=R\, A^i\, R^{-1}\) for all \(i\), then \((\widetilde{(A^{[L]})}^{\, U^{[L]}}_g)^I=R\, (A^{[L]})^I\, R^{-1}\) for every blocked index \(I\).

The twisted tensor satisfies the composition law

i.e. twisting by \(gh\) is the same as first twisting by \(h\) and then by \(g\).

Twisting by the identity group element is trivial: \(\widetilde{A}_1=A\).

The twisted transfer map of \(A\) is the mixed transfer operator of the pair \((A,B_u)\):

The local intertwining equation gives, for every matrix \(X\),

No peripheral phase is discarded.

For every virtual input \(Z\), the phase-retaining iterate identity gives

This limit allows physical endpoint functionals of any finite support; the one-site physical limit below is its corresponding trace contraction.

If \(u\) is unitary, then replacing Kraus operators \(\{ A_i\} \) by their unitary mixtures \(\{ \sum _j u_{ij} A_j\} \) preserves the channel:

Let \(A\) be the printed AKLT tensor and let \(u=e^{i\pi S_z}\). For every \(L\geq 0\) and \(X\in M_2(\mathbb {C})\),

Thus \(L=1\) is exactly the zero-vector exception. Length zero has coefficient two, and every length at least two has positive squared norm. This makes explicit the admissible lengths in the value \(R_L(u)=1\) stated in  [ 229 , line 399 ] .

The tensor \(B\) is unital and has positive trace-one stationary density \(\Lambda =\mathbb {1}/2\). For every \(N\geq 0\),

Thus the actual unblocked physical endpoints witness string order with limiting absolute value \(4/9\) and a nonscalar physical unitary. The same value holds for the existing representative \(A\) of Definition 15.3.7.

Let \(P,Q\) be physical matrices with \(PQ=-QP\). No normal tensor with \(D\ge 1\) satisfies \(\sum _jP_{ij}A^j=\zeta \, XA^iX^{-1}\) and \(\sum _jQ_{ij}A^j=\eta \, YA^iY^{-1}\) for all \(i\), with \(X,Y\in \mathrm{GL}_{D}(\mathbb {C})\) and \(\zeta ,\eta \neq 0\).

If the length-\(L\) words of the family \((A_i^\dagger )_i\) span \(M_{D}(\mathbb {C})\), then the length-\(L\) words of \(A\) also span \(M_{D}(\mathbb {C})\).

For \(W\in \mathbb {C}^{m\times n}\), a tensor \(C\) with physical dimension \(n\), and \(L\ge 0\), blocking the rotated tensor over \(L\) sites is rotating the blocked tensor by the \(L\)-fold Kronecker power of \(W\):

Let \(A\) be injective, assume \(u u^\dagger = \mathbb {1}\), and let \(\Lambda \) be a positive definite boundary state with \(\operatorname{tr}(\Lambda )=1\) and \(\mathcal{E}_A^\dagger (\Lambda )=\Lambda \). If \(V\) is a unitary and \(\mu \) a phase satisfying, for every physical index \(i\),

then

There are \(C{\gt}0\) and \(0{\lt}r{\lt}1\) such that \(\| \mathcal{E}_A^N-P\| \leq Cr^N\) for every \(N\geq 1\). The norm is the operator norm induced by the row-sum matrix norm. The same rate satisfies the following assertion: for every linear insertion \(F\) there is \(C_F{\gt}0\) such that every \(N\geq 1\) obeys

For the printed cluster tensor \(A^0=|0\rangle \langle +|\), \(A^1=|1\rangle \langle -|\), let \(u=-\sigma _x\), \(V=\sigma _y\) and \(\Lambda =\mathbb {1}/2\). The transfer map is unital, \(\Lambda \) is a positive trace-one stationary density, and \(V\) implements the local symmetry. Every pair of physical indices satisfies \(\operatorname{tr}(V\Lambda A_nA_j^\dagger )=0\). For arbitrary one-site physical endpoints \(x,y\),

Hence no such endpoints give a positive limiting absolute value for this fixed twist. This does not quantify over other possible twists. This scope is recorded in docs/paper-gaps/pgwsvc08_string_order_virtual_boundary.tex.

Let \(A^0=|0\rangle \langle +|\) and \(A^1=|1\rangle \langle -|\) be the printed cluster tensor, and let \(u=-\sigma _x\). The periodic vector is nonzero for every \(L\geq 0\), with

At positive lengths the vector is the cyclic product of controlled-\(Z\) gates applied to \(|+\rangle ^{\otimes L}\). This convention gives \(|-\rangle \) at length one and \(|+\rangle \otimes |+\rangle \) at length two. The overlap is the value stated in  [ 229 , line 411 ] .

With \(\mathcal E_x^{(2)},\mathcal E_y^{(2)}\) the insertions on the two-site endpoint blocks, every single-site middle length \(N\geq 0\) satisfies

Odd middle lengths are included. The two-site blocked tensor also has physical string-order parameter one and satisfies the positive-limit projective predicate with the nonscalar unitary \(u\otimes u\).

If the bond dimension satisfies \(D \ge 1\) and two projective representations \(\rho _1,\rho _2\) (with cocycles \(\omega _1,\omega _2\)) both arise as virtual representations of the same injective MPS tensor \(A\) under an on-site symmetry \(U\), then \(\omega _2\sim \omega _1\), and hence also \(\omega _1\sim \omega _2\) by symmetry.

If \(\omega _1\sim \omega _2\) and \(gh=hg\), then \(\beta _{\omega _1}(g,h)=\beta _{\omega _2}(g,h)\).

If every eigenvalue \(z\) of a Hermitian matrix \(M\) has \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq \delta \), then \(\delta \lVert x\rVert ^2\leq \operatorname {Re}\langle x,Mx\rangle \) for all \(x\perp \ker M\). Conversely, if \(M\geq 0\) satisfies this quadratic-form bound, then every eigenvalue \(z\) of \(M\) has \(\operatorname {Re}z\geq 0\), and \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq \delta \). Let \(0\leq A\leq B\) be matrices with \(\ker A\subseteq \ker B\) and such a gap \(\delta \) for \(A\), and let \(t\geq 0\). Then \(C_t=A+t(B-A)\) is positive semidefinite, \(\ker C_t=\ker A\), and every eigenvalue \(z\) of \(C_t\) satisfies \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq \delta \). If \(\ker A\neq 0\), then \(0\) is an eigenvalue of every \(C_t\).

Let \(x\mapsto A_x\) be a continuous family of one-site injective tensors of nonzero bond dimension, and let \(S\) be a compact parameter set. Then there is \(\delta {\gt}0\) such that, for every \(x\in S\), every \(N\geq 2\) and every \(v\perp \ker H_{N,2}(A_x)\),

This is the uniform-gap conclusion of [ 236 , Appendix A, lines 2575–2580 ] for one-site injective tensors and the two-site parent interaction; the source also treats non-injective normal forms [ 176 ] .

If \(V\) and \(u\) are unitary and the local physical intertwining relation holds, then the transfer map is covariant.

Transfer-map covariance and commutation in the doubled transfer picture are equivalent for any matrix \(V\).

Assume that \(A\) is one-site injective, \(VV^\dagger =\mathbb {1}\), and the transfer map is covariant. Then there is \(u\) with \(uu^\dagger =\mathbb {1}\) satisfying the local physical intertwining relation. Covariance and coisometry of \(V\) alone suffice for the conclusion.

Let \(D{\gt}0\), \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), \(\Lambda {\gt}0\), \(\operatorname{tr}\Lambda =1\), and \(\mathcal{E}_A^*(\Lambda )=\Lambda \). Assume the simple-peripheral purity property: whenever \(X\ne 0\), \(|\lambda |=1\), and \(\mathcal{E}_A(X)=\lambda X\), one has \(\lambda =1\) and \(X\in \mathbb {C}\mathbb {1}\). Let \(H=H^\dagger \), \(H\notin \mathbb {C}\mathbb {1}\), and \([L_H,E]=0\).

There is a nonscalar Hermitian physical generator \(K\) such that \(u(t)=e^{itK}\) is unitary and satisfies (51) with \(V(t)=e^{itH}\) for every real \(t\). There is \(\varepsilon {\gt}0\) such that for every \(0{\lt}|t|{\lt}\varepsilon \), \(u(t)\notin \mathbb {C}\mathbb {1}\) and

In particular, \(A\) has physical string order with Hermitian identity endpoints in the sense of Definition 10.3.1.7.

Let \(H=H^\dagger \) and \([L_H,E]=0\). There is a Hermitian physical matrix \(K\) such that, for every physical index \(i\) and real \(t\),

where \(u(t)=e^{itK}\) and \(V(t)=e^{itH}\). Both paths are continuous unitary representations of \((\mathbb {R},+)\). The construction allows arbitrary linear dependence among the \(A^i\).

The physical operation \(u(t)\) on the open physical leg equals the two virtual operations \(V(t)\) and \(V(t)^\dagger \):

= .

Let \(A\) and \(B\) be normalized MPS tensors. Every eigenvalue \(\mu \) of \(F_{AB}\) satisfies \(|\mu |\leq 1\).

Let \(A\) be an injective MPS tensor and \(B\) a tensor of the same bond dimension. If \(A\) and \(B\) generate the same matrix-product vectors, then there is \(X \in \mathrm{GL}_D(\mathbb {C})\) with \(B^i = X A^i X^{-1}\) for all \(i\). In tensor-network notation the gauge conjugates the local tensor on its virtual legs, with the physical index \(i\) on the upper leg:

Let \(A\) be an injective MPS tensor with \(D\ge 1\) and \(B\) a tensor of the same bond dimension. Suppose that for every \(N\ge 1\) there is \(c_N\in \mathbb {C}\) with \(|c_N|=1\) and \(|V^{(N)}(B)\rangle =c_N|V^{(N)}(A)\rangle \). Then there are \(\lambda \in \mathbb {C}\) with \(|\lambda |=1\) and \(X\in \mathrm{GL}_D(\mathbb {C})\) with \(B^i=\lambda \, XA^iX^{-1}\) for all \(i\), and \(c_N=\lambda ^N\). This is the single-block case of the Fundamental Theorem of matrix product vectors [ 13 , Theorem 2.10 ] , with unimodular proportionality constants.

If \(A\) and \(B\) are gauge equivalent, then they generate the same MPV family.

Let \(A\) be an injective tensor, symmetric under a unitary representation \(U\). Assume \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), \(\Lambda \) is positive definite, \(\operatorname{tr}(\Lambda )=1\), and \(\mathcal{E}_A^\dagger (\Lambda )=\Lambda \). Then the virtual-boundary nondecay predicate holds for every group element \(g\).

Let \(D{\gt}0\), let \(\mathcal{E}_A\) be irreducible and peripheral-primitive, and suppose \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), \(\Lambda {\gt}0\), \(\operatorname{tr}\Lambda =1\), and \(\mathcal{E}_A^*(\Lambda )=\Lambda \). If \(V\) is unitary and \(\sum _j u_{ij}A^j=VA^iV^\dagger \) for every \(i\), then the complex correlator itself converges:

No additional invariance assumption on \(\Lambda \) is required.

If \(A\) is an injective MPS tensor, then its transfer map \(\mathcal{E}_A\) is irreducible.

Under the preceding irreducible unital hypotheses, \(\rho (\mathcal{E}_u)\leq 1\). Equality holds exactly when there are a unitary \(V\) and a phase \(\mu \), \(|\mu |=1\), such that

With the canonical dual density this is precisely virtual local covariance including boundary invariance. This statement concerns the actual spectral radius, not just a bound on a supplied eigenpair.

If \(D{\gt}0\) and \(\rho :G\to \mathrm{GL}_D(\mathbb {C})\) obeys \(\rho (g)\rho (h)=\omega (g,h)\rho (gh)\), then \(\omega \) is a \(\mathbb {C}^\times \)-valued \(2\)-cocycle.

Let \(P\) satisfy \(P\bar P=-\mathbb {1}\), as \(\sigma _y\) does. No normal tensor with \(D\ge 1\) admits \(X\in \mathrm{GL}_{D}(\mathbb {C})\) and \(\zeta \in \mathbb {C}\) with \(\sum _jP_{ij}\bar A^j=\zeta \, X^{-1}A^iX\) for all \(i\). In particular no normal spin-\(\tfrac 12\) tensor carries the Wigner time reversal \(A^i\mapsto \sum _j(\sigma _y)_{ij}\bar A^j\) in this form [ 17 , Section III.A ] .

No injective spin-\(\tfrac 12\) tensor with \(D\ge 1\) is invariant up to a phase under the Wigner time reversal: it is not the case that for every \(N\ge 1\) the matrix product vector of \(\tilde A^i=\sum _j(\sigma _y)_{ij}\bar A^j\) is \(c_N|\psi _N(A)\rangle \) with \(|c_N|=1\) [ 17 , Section III.A ] .

Every complex matrix satisfies \(M=VP\) and \(V^\dagger V=\Pi \).

If \(B\) is injective, then \(\widehat B\colon \mathbb {C}^{D^2}\to \mathbb {C}^n\) is injective.

Let \(\sigma \) be a fixed point of the adjoint transfer map \(\sum _i (A^i)^\dagger \sigma A^i = \sigma \), and let \(S\) be invertible with \(S^\dagger S = \sigma \). Define \(A'^i = SA^iS^{-1}\). Then \(\sum _i (A'^i)^\dagger A'^i = \mathbb {1}\), i.e. the gauged Kraus map is trace-preserving. This is the left-canonical normalization.

Let \(A\) and \(B\) be gauge equivalent. For every \(L\) one has \(G_L(A)=G_L(B)\).

Let \(\zeta \neq 0\). For every \(L\) one has \(G_L(\zeta A)=G_L(A)\).

Let \(A\) be injective, and assume \(u u^\dagger = \mathbb {1}\), \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), \(\Lambda \) is positive definite, \(\operatorname{tr}(\Lambda )=1\), and \(\mathcal{E}_A^\dagger (\Lambda )=\Lambda \). Then virtual local covariance holds if and only if there exist a unitary \(V \in M_{D}(\mathbb {C})\) and a phase \(\mu \) with \(|\mu |=1\) such that \(\mathcal{E}_u(V) = \mu V\). By Theorem 10.3.2.17, this witness form is equivalent to the peripheral-spectrum condition \(\rho (\mathcal{E}_u)=1\).

Let \(A\) and \(B\) be irreducible trace-preserving normalized MPS tensors of bond dimension \(D\). If \(\rho _{\operatorname{spec}}(F_{AB})\geq 1\), then \(A\) and \(B\) are gauge-phase equivalent.

Let \(E:H\to H'\) be an isometry and \(B\) a Hermitian two-site matrix. Put \(\widetilde B=(E\otimes E)B(E\otimes E)^\dagger \). Suppose its periodic translates commute on a ring of length \(N\ge 2\). Their complete product is supported on the included one-site space at every site, and

This is the isometric product identity used in the physical-support passage of [ 13 , Appendix C.2 ] .

For an arbitrary finite family of square matrices, isometric physical mixing preserves its transfer map, injectivity, and left-canonical form. Arbitrary physical mixing preserves a common virtual gauge. Let \(\mathcal K=\{ \mathcal K^{pq}\} _{p,q=0}^{d-1}\) be an MPO tensor and set \(\mathcal K_V^{ij}=\sum _{p,q}V_{i,p}\overline{V_{j,q}}\mathcal K^{pq}\). Write \(A^{(p,q)}=\mathcal K^{pq}\) and \(A_V^{(i,j)}=\mathcal K_V^{ij}\) for the associated doubled-index MPS tensors. For every \(V\), the matrix in (24) satisfies \(A_V^\tau =\sum _\sigma W_{\tau ,\sigma }A^\sigma \).

Suppose now that \(V^\dagger V=\mathbb {1}\). Then

Here \(\Pi _{\mathcal K}\) is the orthogonal projection onto the joint column span of the physical slices of \(\mathcal K\). Left-canonical normalization of \(A\) implies left-canonical normalization of \(A_V\).

For arbitrary \(V\), positivity of every positive-length periodic operator for \(\mathcal K\) implies the same positivity for \(\mathcal K_V\). Moreover, if two doubled-index tensors are related by one virtual gauge, applying the same \(V\) to their physical indices preserves that gauge.

This isometric invariance applies to the normal-block and per-sector constructions in [ 13 , lines 217–246, 1628–1665, 1740–1782 ] .

Let \(T\) be a matrix product operator tensor, \(A\) a matrix product state tensor, and \(B\) an injective matrix product state tensor of positive bond dimension, all with the same physical alphabet. Suppose that for every \(N\geq 1\)

Then there is \(\lambda \in \mathbb {C}\) with \(|\lambda |=1\) and \(c_N=\lambda ^N\) for every \(N\geq 1\). In particular this holds when \(T\) is a matrix product unitary and \(A\) is injective, as in  [ 200 , Section II.B ] .

Let \(G\) be finite, and give \(U(1)\) and \(\mathbb {C}^\times \) the trivial \(G\)-actions. Inclusion of the unit circle induces an equivalence

More concretely, for a \(\mathbb {C}^\times \)-valued cocycle \(\omega \), define its pointwise phase and radial residual by

Then \(u\) is a cocycle, \(r\) is a positive coboundary, and hence \(\omega \) is cohomologous to the inclusion of \(u\). The inclusion map is injective for every group because pointwise phase is its left inverse; finiteness is used only to prove surjectivity.

This is an algebraic group-cohomology statement for a finite discrete group. The footnote at [ 197 , line 1931 ] instead invokes a comparison of sheaf cohomology with topological coefficient groups; the distinction and the direct finite-group proof are recorded in [ 38 ] .

If \(G\) is finite and \(\omega \) is a cocycle, then

If \(G\) is finite, the matrices \(L_g^\phi \) and \(R_g^\phi \) are unitary for every \(g\in G\).

If a commuting pair \(g,h\) has \(\beta _\omega (g,h)\neq 1\), then \(\omega \) has a non-trivial class.

Let \(D{\gt}0\) and let \(A\) be a normal tensor. Then \(A^i\neq 0\) for at least one \(i\).

Let \(A\) and \(B\) be irreducible MPS tensors of the same bond dimension with \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\) and \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\). If \(\| \langle V^{(N)}(A) | V^{(N)}(B) \rangle \| \to 1\), then \(A\) and \(B\) are gauge-phase equivalent.

If \(A\) has MPV-family invariance under \(U\), then \(A^{[L]}\) has MPV-family invariance under \(U^{[L]}\) for every \(L\).

For tensors \(A\) and \(B\) of bond dimension \(D\geq 1\),

Let \(A,u\) be arbitrary and suppose a unitary \(V\) and \(\mu \in \mathbb {C}\) satisfy \(\sum _j u_{ij}A_j=\mu V A_iV^\dagger \) for every physical index \(i\). At every length with \(\Phi _L(A)\ne 0\),

The numerator identity \(\operatorname {Tr}(\mathcal{E}_u^L)=\mu ^L\operatorname {Tr}(\mathcal{E}_A^L)\) holds also at zero periodic vectors, without purity or normalization.

Under the same canonical purity assumptions, fix a support length \(k\) and a physical operator \(O\) on those \(k\) sites. Then

In particular, for fixed endpoint blocks \(x,y\) of lengths \(p,q\) and fixed middle length \(N\), the limit for \(x\otimes u^{\otimes N}\otimes y\) is \(\operatorname{tr}[\Lambda \mathcal{E}_x\mathcal{E}_u^N\mathcal{E}_y(\mathbb {1})]\). The case \(p=q=1\) is display SOPMP of [ 229 , lines 176–181 ] . The middle length is fixed before the exterior \(n\) tends to infinity; no exchange of limits or rate uniform in \(N\) is asserted.

Assume the faithful canonical purity hypotheses stated before Theorem 10.3.1.2. There exist \(C_0{\gt}0\), \(0{\lt}r{\lt}1\), and an integer \(N_0\geq 1\) such that, for \(L\geq N_0\), the periodic vector is nonzero, \(|Z_L|\geq 1/2\), and \(|Z_L-1|\leq C_0r^L\). For each fixed support \(k\) and physical matrix \(O\) there is \(C_O{\gt}0\) such that every exterior length \(n\geq N_0\) satisfies

In addition, for every \(q{\gt}\rho (\mathcal{E}_u)\) there are \(C_u{\gt}0\) and \(L_0\) such that \(|\langle u^{\otimes L}\rangle _L|\leq C_u q^L\) whenever \(L\geq L_0\); it decays when \(q{\lt}1\). No prefactor uniform over growing supports is asserted.

Under these canonical purity assumptions, \(Z_L\to 1\) as \(L\to \infty \). In particular, \(\Phi _L(A)\ne 0\) for all sufficiently large \(L\). No nonvanishing assertion at every short length is required.

Assume the canonical purity conditions above and let \(u\) be unitary. Either \(\rho (\mathcal{E}_u){\lt}1\) and the complex full-ring expectation tends to zero, or \(\rho (\mathcal{E}_u)=1\) and its magnitude tends to one. In the latter case, choose the unitary intertwiner and phase \(\mu \) supplied by the twisted spectral criterion. The expectation is eventually exactly \(\mu ^L\), so its phase-adjusted value tends to one; if \(\mu \ne 1\), the unadjusted complex sequence has no limit. This is the phase qualification of display RL, lines 381–388, recorded in  [ 183 ] . No implication concerning a parameterized Hamiltonian gap is asserted.

For every tensor \(A\), physical matrix \(u\), and length \(L\), the normalized full-ring expectation is

Both sides use the zero convention at a vanishing periodic vector.

Theorem 1 of  [ 229 ] states the following with the literal source condition \(u\ne \mathbb {1}\) on the string-order side. A pure finitely correlated state has such string order if and only if there exist a unitary \(\widetilde u\ne \mathbb {1}\), a virtual matrix \(V\), and physical indices \(n,m\) such that

This printed assertion is distinct from the explicitly projective convention in Definition 10.3.1.7. The literal assertion is false: Example 10.3.9.1 satisfies its string-order condition but has no nonidentity fixed twist. The scalar-phase obstruction and the explicit counterexample are recorded in  [ 100 ] .

This is Theorem 2 of  [ 229 ] for a fixed physical unitary \(u\). Suppose \(\Lambda {\gt}0\), \(\operatorname{tr}\Lambda =1\), \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), and \(\mathcal{E}_A^*(\Lambda )=\Lambda \). Assume that \(\mathcal{E}_A(X)=\lambda X\), \(X\ne 0\), and \(|\lambda |=1\) imply \(\lambda =1\) and \(X\in \mathbb {C}\mathbb {1}\). Then

Scalar unitaries are included in this fixed-\(u\) statement.

Let \(A\) be normal and unital, and let \(\Lambda \) be invertible with \(\mathcal{E}_A^*(\Lambda )=\Lambda \). For all virtual matrices \(X,Y\), there are physical operators \(x,y\) on \(q=D^2\) sites such that, for every physical twist \(u\) and every individual middle length \(N\),

Neither endpoint nor its support depends on \(u\) or \(N\).

Let \(A\) have faithful normalized canonical density \(\Lambda \) and the source’s simple-peripheral purity property. For every unitary \(u\) with \(\rho (\mathcal{E}_u)=1\), there are Hermitian one-site endpoint factors supported on \(D^2\) sites at each end for which

The twist remains the supplied \(u\), and \(N\) counts individual middle sites. No one-site injectivity assumption is imposed. An existential claim of nontrivial physical action separately requires the projective convention of Definition 10.3.1.7.

Assume in addition that \(\operatorname{tr}\Lambda =1\), and let \(V\) be unitary. There are Hermitian one-site matrices \(x_j,y_j\), \(0\le j{\lt}q=D^2\), for which the product observables \(x=\bigotimes _{j=0}^{q-1}x_j\) and \(y=\bigotimes _{j=0}^{q-1}y_j\) satisfy

Let \(A\) be normal and let \(\Omega \in M_{D}(\mathbb {C})\) be invertible with \(\mathcal{E}_A(\Omega )=\Omega \). Every \(Y\in M_{D}(\mathbb {C})\) has a physical representative \(O\) on \(q=D^2\) sites satisfying \(\mathcal{E}_O^{[q]}(\Omega )=Y\). In particular, when \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), every right virtual boundary is \(\mathcal{E}_O^{[q]}(\mathbb {1})\) for such an \(O\).

Let \(M,N\) belong to the span of the length-\(L\) words of a tensor \(A\). There is an operator \(O\) on \(L\) physical sites whose inserted transfer map satisfies \(\mathcal{E}_O(X)=MXN^\dagger \) for every virtual matrix \(X\). This is the two-sided multiplication used in the observable choices of [ 14 , Appendix C ] and [ 219 , Lemma 2 ] .

Let \(\Lambda :\mathbb C^d\to \mathbb C^m\) be a linear map. For every word \(\sigma \) of length \(N\) in the \(m\)-letter alphabet,

where \(\tau \) ranges over words in the \(d\)-letter alphabet.

For the fixed data above,

This is a criterion for the fixed physical correlator, not a replacement by arbitrary virtual boundary matrices.

For the same fixed unitary twist and its peripheral intertwiner, physical one-site endpoints with positive limiting magnitude exist exactly when

Matrix units realize the left coefficient; the pairing identity supplies the other endpoint. The source-facing version assumes the canonical spectral-purity criterion, deriving irreducibility rather than imposing one-site injectivity.

This does not assert that \(u_0\ne \mathbb {1}\), or classify scalar phases as nontrivial physical actions. The existential Theorem 1 over twists and the physical reduced-state symmetry Theorem 2 remain separate.

Let \(A,\Lambda ,x,y,u\) be arbitrary and choose a real rate \(r{\gt}\rho (\mathcal{E}_u)\). There is a constant \(C{\gt}0\) such that every middle length \(N\geq 0\) satisfies \(|S_N(x,y,u)|\leq Cr^N\). When \(r{\lt}1\), this gives the exponential decay asserted in [ 229 , lines 241–243 ] .

Under the preceding canonical assumptions, let a unitary \(V\) and \(|\mu |=1\) satisfy \(\sum _j u_{ij}A_j=\mu V A_iV^\dagger \). There is \(0{\lt}r{\lt}1\) such that, for every pair of fixed physical endpoints \(x,y\), some \(C_{x,y}{\gt}0\) bounds all positive middle lengths by

This is the quantitative, phase-retaining form of [ 229 , lines 249–255 ] .

Suppose \(A\) has faithful normalized canonical density and the source’s simple-peripheral purity property. For a fixed unitary \(u\), physical one-site endpoints with positive limiting string magnitude exist if and only if there are a virtual matrix \(V\), a phase \(|\mu |=1\), and physical indices \(n,m\) satisfying

The physical twist is unchanged. The nonzero coefficient excludes \(V=0\), and peripheral rigidity identifies its direction with that of a unitary intertwiner, so the fixed-intertwiner endpoint criterion applies.

For fixed physical endpoint matrices \(x,y\), set

The exact identity \(\mathcal{E}_u^N(X)=\mu ^N V\mathcal{E}_A^N(V^\dagger X)\) gives

Thus \(S_N=\mu ^N\alpha _y\beta _x+o(1)\). The complex sequence \(S_N\) itself need not converge. These are the actual physical endpoints in the existing transfer-form correlator.

Let \(A\) have the faithful normalized canonical density and the simple-peripheral purity property of Lemma 10.3.1.9. Then physical string order in the projective sense is equivalent to the existence of a nonscalar unitary \(u_0\), a virtual matrix \(V\), and physical indices \(n,m\) such that

This is the explicitly corrected criterion, rather than the unchanged printed assertion above.

Let \(\rho _1,\rho _2\) be projective representations with factor systems \(\omega _1,\omega _2\). Then

Let \(A\) be an injective MPS tensor. If \(\rho ,\sigma \geq 0\) are nonzero fixed points of \(\mathcal{E}_A\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Let \(A\) and \(B\) be irreducible trace-preserving tensors of positive, possibly different, bond dimensions. If \(|O_{AB}(N)|\longrightarrow 1\), then their bond dimensions are equal.

Let \(A\) be normal with \(D\ge 1\) and let \(X\) be unitary with \((A^i)^{T}=\zeta \, X^\dagger A^iX\) for all \(i\). Then \(X\bar X=\pm \mathbb {1}\), and the sign is the same for every unitary \(X\) with this property [ 17 , Section III.A ] .

For a physical matrix \(K_0\) and a virtual matrix \(V\),

If \(H=H^\dagger \), then \(L_H=L_H^\dagger \) and \((L_HF)_{(a,b),i}=(HA^i-A^iH)_{ab}\). Hence \(FK_0=L_HF\) implies \(\sum _j(K_0^{\top })_{ij}A^j=HA^i-A^iH\) for every \(i\).

The Choi Gram matrix \(G=FF^\dagger \) and the transfer matrix \(E\) have different index pairings. For all virtual indices \(a,b,c,e\),

If \(H=H^\dagger \), their commutators satisfy

Consequently \([L_H,E]=0\) if and only if \([L_H,FF^\dagger ]=0\).

Suppose \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), \(\mathcal{E}_A^*(\Lambda )=\Lambda \), \(\operatorname{tr}\Lambda =1\), and every fixed matrix of \(\mathcal{E}_A\) is scalar. If \(HA^i-A^iH=cA^i\) for every \(i\), then \(c=0\) and \(H\in \mathbb {C}\mathbb {1}\). Neither \(H\) nor \(\Lambda \) needs to be Hermitian for this assertion.

Suppose \(C\) is one-site injective and \(\sum _j U_{ij}C^j=VC^iV^\dagger \) for every \(i\), with \(U\in M_{d}(\mathbb {C})\) and \(V\in M_{D}(\mathbb {C})\) arbitrary. Then \(\beta _V=\Phi _C U^{\mathsf T}\Psi _C\).

For \(V,X\in M_{D}(\mathbb {C})\), one has \(\beta _V(e_DX)=e_D(VXV^\dagger )\).

For any finite-dimensional tensor \(B\), the complex dimensions of the fixed spaces of \(\mathcal{E}_B\) and its trace adjoint \(\mathcal{E}_B^*\) agree: \(\dim _{\mathbb {C}}\ker (\mathbb {1}-\mathcal{E}_B^*)=\dim _{\mathbb {C}}\ker (\mathcal{E}_B-\mathbb {1})\). Positivity, normalization, irreducibility and positive bond dimension are not required. This is a finite-dimensional auxiliary identity for [ 236 , Appendix C, lines 2653–2717 ] .

Let \(B\) be a tensor and let \(\sigma \geq 0\) satisfy \(\mathcal E_B^*(\sigma )=\sigma \). Let \(K:\mathbb {C}^D\to \mathbb {C}^k\) be an isometry onto the support of \(\sigma \), and put \(A^i=K^\dagger B^iK\). Then, for every \(D\times D\) matrix \(Z\),

Neither unitality nor normalization of \(\sigma \) is required.

Suppose a unitary physical group representation \(U\) and a unitary projective virtual representation \(V\), with factor \(\omega \), satisfy \(\sum _jU(g)_{ij}A^j=V(g^{-1})A^iV(g^{-1})^\dagger \). For every \(L\geq 0\), the same \(V\) implements the covariance of \(A^{[L]}\) under \(U^{[L]}\). In particular, its factor remains exactly \(\omega \). Source context: [ 236 , Appendix C, lines 2653–2717 ] .

Let \(T\) be first countable. At each \(t\in T\), let \(A_t\) be an irreducible unital tensor of positive bond dimension \(D(t)\le M\). Suppose that fixed ambient tensors \(Q_t\) represent their positive-length periodic rays and vary continuously at \(t_0\), and that \(A_{t_0}\) is injective. Under one exact physical unitary covariance, with pointwise unitary virtual representatives, assume a common bound \(0\le q{\lt}1\) for every nonunit transfer eigenvalue. Then the virtual factor systems are cohomologous to that at \(t_0\) throughout some neighborhood of \(t_0\).

Let \(T\) be first countable, and let normal unital tensors \(A_t\) have bounded bond dimensions \(D(t)\le M\). Suppose that fixed ambient tensors \(Q_t\) represent their positive-length periodic rays and vary continuously at \(t_0\). Assume exact covariance under a common physical unitary action, pointwise unitary virtual representatives, and a common bound \(0\le q{\lt}1\) on all transfer eigenvalues different from \(1\). Then their virtual factor systems are locally cohomologous to that at \(t_0\). Neither minimal tensors nor stationary densities need be chosen continuously. The uniform transfer bound is a supplied hypothesis.

For fixed \(d,D\), the map \(C\mapsto \Phi _C\) is continuous in the operator norm.

Let \(U\in M_{d}(\mathbb {C})\) and \(V\in M_{D}(\mathbb {C})\) satisfy \(\sum _j U_{ij}C^j=VC^iV^\dagger \) for every \(i\). Then \(\Phi _C U^{\mathsf T}=\beta _V\Phi _C\). Neither matrix is required to be unitary or invertible.

For \(x\in \mathbb {C}^d\), one has \(\Phi _Cx=e_D(\sum _i x_i C^i)\).

If \(\operatorname{span}_{\mathbb {C}}\{ C^i\} =M_{D}(\mathbb {C})\), the map \(B\mapsto \Psi _B\) is continuous at \(B=C\).

If \(\operatorname{span}_{\mathbb {C}}\{ C^i\} =M_{D}(\mathbb {C})\), then \(\Phi _C\Psi _C=I\).

If \(\operatorname{span}_{\mathbb {C}}\{ C^i\} =M_{D}(\mathbb {C})\), then \(\Phi _C\) is surjective.

Let \(A\) be an injective unital tensor, and let \(B_n\) be unital irreducible tensors of one fixed bond dimension. Suppose that their periodic rays are represented by tensors \(Q_n\) in one fixed ambient dimension, with \(Q_n\to Q_0\), and that \(Q_0\) has the same positive-length periodic rays as \(A\). Assume \(0\le q{\lt}1\), and that every transfer eigenvalue different from \(1\), for \(A\) and all \(B_n\), has modulus at most \(q\). Suppose that one physical unitary action has exact inverse-index virtual covariance on these tensors, with pointwise unitary projective representatives having factor systems \(\omega _0\) and \(\omega _n\). Then \(\omega _n\) is cohomologous to \(\omega _0\) for all sufficiently large \(n\).

For fixed \(d,D\geq 0\), the set of tensors \(A=(A^i)_{i=1}^d\) satisfying \(\sum _i A^i(A^i)^\dagger =I_D\) is compact in the topology of their matrix entries. No injectivity or stationary-density assumption is needed.

Let \(A\) be a unital injective tensor with positive bond dimension \(D\). Let \(B_n\) be unital tensors of a fixed bond dimension \(k\), converging to a unital tensor \(b\) with \(\dim _{\mathbb {C}}\ker (\mathcal{E}_b-\mathbb {1})=1\). Suppose \(B_n\) has a positive-definite adjoint stationary matrix \(\sigma _n\) of trace one. Let \(\tau \) be a positive semidefinite adjoint stationary matrix for \(b\), with trace one, and let \(K:\mathbb {C}^E\to \mathbb {C}^k\) be an isometry whose range is the support of \(\tau \). Write \(C^i=K^\dagger b^iK\).

Suppose \(0\leq q{\lt}1\) bounds the modulus of every nonunit eigenvalue of both \(\mathcal{E}_A\) and \(\mathcal{E}_b\). Suppose also that the periodic coefficient spans of \(A\) and \(b\) agree for all sufficiently large lengths. Let the same unitary physical group representation \(U\) act covariantly on \(A\) and every \(B_n\), through pointwise unitary projective virtual representations with respective factors \(\omega _0\) and \(\omega _n\): \(\sum _j U(g)_{ij}A^j=V_0(g^{-1})A^iV_0(g^{-1})^\dagger \), and likewise for \(B_n\) and \(V_n\). Then \(\omega _n\sim \omega _0\) for all sufficiently large \(n\). No continuity of the virtual representations or of \(\sigma _n\) is required. The transfer bound and periodic-span equality are supplied hypotheses; this is an auxiliary reduction for [ 236 , Appendix C, lines 2653–2717 ] .

Let \(P\) be a symmetric gapped interaction path on the unit interval, with an independent continuous exact MPS realization \(Q\) of its unique finite-chain ground lines. Suppose \(0\le q{\lt}1\) has the following property: at every parameter, every positive-dimensional unital injective tensor with the same positive-length periodic rays as \(Q\) has all transfer eigenvalues different from \(1\) of modulus at most \(q\).

There are a common unit-modulus physical character \(\chi \), pointwise unital injective tensors \(A_t\) of positive bond dimensions bounded by the ambient dimension of \(Q\), and pointwise unitary projective virtual representations with factor systems \(\omega _t\), such that \(A_t\) has the same positive-length periodic rays as \(Q_t\), the physical action \(\chi (g)^{-1}U_g\) has exact inverse-index virtual covariance on \(A_t\), and \(\omega _s\) and \(\omega _t\) are cohomologous for all \(s,t\in [0,1]\). No continuous choice of the tensors, stationary densities, or virtual matrices is required. The uniform transfer bound is supplied explicitly; this statement does not derive it from the Hamiltonian gap.

Let \(f_n\to f_\infty \) in operator norm on a finite-dimensional complex normed space \(V\). Suppose \(v\neq 0\), \(f_n v=v\) for every \(n\), and the eigenvalue \(1\) has algebraic multiplicity one for every \(f_n\). If there exists \(q{\lt}1\) such that every eigenvalue \(\lambda \neq 1\) of every \(f_n\) satisfies \(|\lambda |\leq q\), then \(\ker (f_\infty -I)=\operatorname{span}_{\mathbb {C}}\{ v\} \). The common spectral bound is an assumption of this statement.

Let \(A,B\) have the same positive bond dimension, with \(A\) injective, and suppose \(B^i=cXA^iX^{-1}\) for \(c\ne 0\) and \(X\) invertible. Let a physical group representation \(U\) have virtual projective representations \(V_A,V_B\) with factors \(\omega _A,\omega _B\) satisfying \(\sum _jU(g)_{ij}A^j=V_A(g^{-1})A^iV_A(g^{-1})^{-1}\) and the corresponding identity for \(B\). Then \(\omega _B\sim \omega _A\). Source context: [ 236 , Section II.F.2 and Appendix C ] .

Let \(T\) be a topological space, let \(t_0\in T\), and let \(C:T\to (M_{D}(\mathbb {C}))^d\) be continuous at \(t_0\), with \(D{\gt}0\) and \(C(t_0)\) one-site injective. Fix a group \(G\) and matrices \(U_g\in \mathrm U(d)\). For each \(t\), let \(\rho _t\) be a unitary projective representation with factor \(\omega _t\), and suppose

Then \(\omega _t\) is cohomologous to \(\omega _{t_0}\) throughout some neighborhood of \(t_0\). No continuity of \(\rho _t\) or \(\omega _t\), injectivity away from \(t_0\), or connectedness of \(T\) is assumed.

Let \(f_n\to f_\infty \) in operator norm on a finite-dimensional complex normed space. Suppose \(v\neq 0\), \(f_n v=v\) for every \(n\), and the eigenvalue \(1\) has algebraic multiplicity one for every \(f_n\). Let \(q\in \mathbb R\) satisfy \(|\lambda |\leq q\) for every eigenvalue \(\lambda \neq 1\) of every \(f_n\). Then every eigenvalue \(\lambda \neq 1\) of \(f_\infty \) also satisfies \(|\lambda |\leq q\). Neither \(q{\lt}1\) nor algebraic simplicity of \(1\) at the limit is required.

Let \(B\) have arbitrary finite bond dimension and \(\dim _{\mathbb {C}}\ker (\mathcal{E}_B-\mathbb {1})=1\). If \(\mathcal{E}_B^*(\sigma )=\sigma \), \(\mathcal{E}_B^*(Z)=Z\), and \(\operatorname{tr}\sigma =\operatorname{tr}Z=1\), then \(Z=\sigma \). Positivity and unitality are not required. This is an auxiliary identity for [ 236 , Appendix C, lines 2653–2717 ] .

Let \(A,B,C,R\) be tensors with the same physical dimension and arbitrary finite bond dimensions. Suppose there is \(M\in \mathbb {R}\) such that \(\| |V^{(N)}(A)\rangle \| \leq M\) for all sufficiently large \(N\), and suppose \(|V^{(N)}(B)\rangle =|V^{(N)}(C)\rangle +|V^{(N)}(R)\rangle \) for all sufficiently large \(N\). If \(\| |V^{(N)}(R)\rangle \| \to 0\), then \(O_{AB}(N)-O_{AC}(N)\to 0\). This is an auxiliary asymptotic statement for the varying-support argument of [ 236 , Appendix C, lines 2653–2717 ] .

Fix a physical dimension \(d\), bond dimensions \(D,E\), and a length \(N\geq 0\). Let \(A_x\to A\) and \(B_x\to B\) along a nontrivial filter, with bond dimensions \(D\) and \(E\), respectively. Suppose that, eventually along this filter, \(\operatorname{span}_{\mathbb {C}}\{ |V^{(N)}(A_x)\rangle \} =\operatorname{span}_{\mathbb {C}}\{ |V^{(N)}(B_x)\rangle \} \). If both limiting periodic vectors are nonzero, then \(\operatorname{span}_{\mathbb {C}}\{ |V^{(N)}(A)\rangle \} =\operatorname{span}_{\mathbb {C}}\{ |V^{(N)}(B)\rangle \} \).

Let \(A,B\) have the same physical dimension and arbitrary bond dimensions. Suppose their periodic coefficient spans agree for all sufficiently large \(N\), and both periodic state norms tend to one. Then \(|O_{AB}(N)|\to 1\).

Let \(A\) be unital with positive bond dimension \(D\) and \(\dim _{\mathbb {C}}\ker (\mathcal{E}_A-\mathbb {1})=1\). Suppose \(0\leq q{\lt}1\) and \(|z|\leq q\) for every \(z\in \operatorname {spec}(\mathcal{E}_A)\) with \(z\ne 1\). Then \(\| |V^{(N)}(A)\rangle \| \to 1\). The spectral bound is an explicit auxiliary assumption in the setting of [ 236 , Appendix C, lines 2653–2717 ] .

Let \(A,B\) have the same physical dimension and arbitrary bond dimensions. At any \(N\geq 0\), equality of the spans of their periodic coefficient vectors implies \(|O_{AB}(N)|=\| |V^{(N)}(A)\rangle \| \, \| |V^{(N)}(B)\rangle \| \). Both vectors are allowed to vanish.

Suppose \(D_0,D_1{\gt}0\) and both projective representations are unitary. For every \(g,h\in G\) and matrix \(M\) on the intertwiner space, \(S_g(S_hMS_h^\dagger )S_g^\dagger =S_{gh}MS_{gh}^\dagger \). The two factor systems need not agree.

If \(\rho _a\) has factor \(\omega _a\), then \(S_gS_h=\overline{\omega _0(g,h)}\omega _1(g,h)S_{gh}\) for all \(g,h\). Unitarity is not required.

If all matrices of \(\rho _0\) and \(\rho _1\) are unitary, then every \(S_g\) is unitary. The dimensions \(D_0,D_1\) may be zero.

Let \(V\) be a finite-dimensional complex normed space, let \(f_n\to f_\infty \) in operator norm, and let \(W\subseteq V\) be invariant under every \(f_n\) and under \(f_\infty \). In any fixed basis of \(V/W\), the matrices of the induced quotient operators converge to the matrix of the quotient operator induced by \(f_\infty \).

Let \(A,B,C,R\) have the same physical dimension, with arbitrary finite bond dimensions for \(B,R\) and positive bond dimensions \(D,E\) for \(A,C\). Suppose \(A,C\) are irreducible and admit positive definite adjoint stationary matrices \(\sigma ,\tau \). Suppose \(\| |V^{(N)}(A)\rangle \| \leq M\) eventually for some \(M\in \mathbb {R}\), and \(|V^{(N)}(B)\rangle =|V^{(N)}(C)\rangle +|V^{(N)}(R)\rangle \) eventually. If \(\| |V^{(N)}(R)\rangle \| \to 0\) and \(|O_{AB}(N)|\to 1\), then \(D=E\). After this identification there exist \(X\in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\setminus \{ 0\} \) such that \(C^i=\zeta XA^iX^{-1}\) for every \(i\). This is a conditional auxiliary reduction for [ 236 , Appendix C, lines 2653–2717 ] , using [ 13 , Lemma equalMPS ] .

Let \(A,B,C,R\) have the same physical dimension and arbitrary finite bond dimensions. Suppose \(\| |V^{(N)}(A)\rangle \| \leq M\) eventually for some \(M\in \mathbb {R}\), and \(|V^{(N)}(B)\rangle =|V^{(N)}(C)\rangle +|V^{(N)}(R)\rangle \) eventually. Assume \(\| |V^{(N)}(R)\rangle \| \to 0\). Convergence \(|O_{AB}(N)|\to 1\) then implies \(|O_{AC}(N)|\to 1\). Neither positivity nor a relation between the bond dimensions is required; the decomposition need not hold at length zero.

Let \(A,B\) have positive bond dimension \(D\) and finite bond dimension \(k\), respectively, with the same physical dimension. Suppose \(A\) is irreducible and has a positive definite adjoint stationary matrix \(\tau \). Suppose \(B\) is unital and \(\dim _{\mathbb {C}}\ker (\mathcal{E}_B-\mathbb {1})=1\). Let \(\sigma \geq 0\) satisfy \(\operatorname{tr}\sigma =1\) and \(\mathcal{E}_B^*(\sigma )=\sigma \). Let \(K:\mathbb {C}^E\to \mathbb {C}^k\) be an isometry with \(KK^\dagger =\operatorname {supp}\sigma \), and put \(C^i=K^\dagger B^iK\). Suppose \(0\leq q{\lt}1\) and every \(z\in \operatorname {spec}(\mathcal{E}_B)\) with \(z\ne 1\) satisfies \(|z|\leq q\). If \(\| |V^{(N)}(A)\rangle \| \leq M\) eventually for some \(M\in \mathbb {R}\) and \(|O_{AB}(N)|\to 1\), then \(C\) is normal and \(D=E\). After identifying the bond dimensions, there exist \(X\in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\setminus \{ 0\} \) with \(C^i=\zeta XA^iX^{-1}\) for every \(i\). The spectral bound and overlap limit are explicit assumptions in this auxiliary reduction for [ 236 , Appendix C, lines 2653–2717 ] .

Let \(B\) be unital, and let \(\sigma \geq 0\) be its unique trace-one complex adjoint stationary matrix. Let \(K:\mathbb {C}^D\to \mathbb {C}^k\) be an isometry onto the entire support of \(\sigma \), and put \(A^i=K^\dagger B^iK\). Suppose \(q{\lt}1\) and every \(\lambda \in \sigma (\mathcal E_B)\) with \(\lambda \neq 1\) satisfies \(|\lambda |\leq q\). Then \(A\) is a normal tensor: its Kraus family is irreducible, \(r(\mathcal E_A)=1\), and its unit-modulus spectrum is \(\{ 1\} \). The transfer bound is an explicit hypothesis. This auxiliary consequence uses the normal-tensor definition of [ 13 , lines 231–235 ] in the support setting of [ 236 , Appendix C, lines 2653–2717 ] .

Let \(B\) be a unital tensor with bond dimension \(k\). Suppose \(\sigma \geq 0\), \(\operatorname{tr}\sigma =1\), and \(\mathcal E_B^*(\sigma )=\sigma \), and suppose \(\sigma \) is the unique trace-one complex matrix fixed by \(\mathcal E_B^*\). Let \(K:\mathbb {C}^D\to \mathbb {C}^k\) be an isometry satisfying \(KK^\dagger =\operatorname {supp}\sigma \), and put \(A^i=K^\dagger B^iK\) and \(\rho =K^\dagger \sigma K\). Then \(A\) is unital, \(\mathcal E_A\) is irreducible, and \(\rho {\gt}0\), \(\operatorname{tr}\rho =1\), and \(\mathcal E_A^*(\rho )=\rho \). In particular, \(D{\gt}0\) follows from these hypotheses. This is an auxiliary preparation result for [ 236 , Appendix C, lines 2653–2717 ] , using [ 244 , Lemma 6.4 and Theorem 6.3 ] .

Let \(B\) be a tensor with bond dimension \(k\), and suppose \(\sigma \geq 0\), \(\operatorname{tr}\sigma =1\), and \(\mathcal E_B^*(\sigma )=\sigma \). Suppose \(\sigma \) is the unique trace-one matrix fixed by \(\mathcal E_B^*\), with uniqueness among all such complex matrices, not merely positive semidefinite ones. Let \(K:\mathbb {C}^D\to \mathbb {C}^k\) be an isometry with \(D{\gt}0\) and \(KK^\dagger =\operatorname {supp}\sigma \), and put \(A^i=K^\dagger B^iK\). If \(A\) is unital, then its primal transfer map \(\mathcal E_A\) is irreducible. This is an auxiliary preparation result for [ 236 , Appendix C, lines 2653–2717 ] , using the channel criterion of [ 244 , Theorem 6.3 ] .

Let \(K:\mathbb {C}^D\to \mathbb {C}^k\) be an isometry, put \(P=KK^\dagger \) and \(Q=\mathbb {1}-P\), and let \(C^i=K^\dagger B^iK\). Suppose \(PB^iQ=0\) for every physical letter \(i\). With \(R^i=QB^iQ\), every \(N{\gt}0\) satisfies \(|V^{(N)}(B)\rangle =|V^{(N)}(C)\rangle +|V^{(N)}(R)\rangle \). No normalization or nonzero bond dimension is required. This is the triangular trace decomposition used in [ 228 , proof of Theorem TIcanonical, lines 785–815 ] , and in [ 236 , Appendix C, lines 2653–2717 ] .

Let \(T\) be a topological space and \(t_0\in T\). Let \(B(t)\) be a unital tensor family of fixed physical dimension \(d\) and ambient bond dimension \(k\), continuous at \(t_0\). Suppose that for every \(t\) there are a positive semidefinite matrix \(\sigma _t\), a dimension \(D_t\), and an isometry \(K_t:\mathbb {C}^{D_t}\to \mathbb {C}^k\) satisfying

Put \(A_t^i=K_t^\dagger B(t)^iK_t\). Let \(U_g\) be a fixed unitary physical representation, and let \(V_{t,g}\) be unitary projective representations on \(\mathbb {C}^{D_t}\), with factor systems \(\omega _t\), such that

If \(A_{t_0}\) is one-site injective and \(\dim \ker (I-\mathcal E_{B(t_0)}^*)=1\), then \([\omega _t]=[\omega _{t_0}]\) for all \(t\) in some neighborhood of \(t_0\). The stationary matrices, full support frames, and virtual representations are not required to be continuous. The ambient tensors and their supported covariance are supplied assumptions; this statement does not construct them from a physical gap.

Let \(C_0\) be one-site injective, let \(\mathcal I\) be an arbitrary set, and let \(U_g\in \mathrm U(d)\) and \(V_g^0\in M_{D}(\mathbb {C})\) satisfy \(\sum _j(U_g)_{ij}C_0^j=V_g^0C_0^i(V_g^0)^\dagger \) for all \(g,i\). For every \(\varepsilon {\gt}0\) there is a neighborhood of \(C_0\) such that, for each tensor \(C\) in that neighborhood, every \(g\in \mathcal I\) and every \(V\in M_{D}(\mathbb {C})\) satisfying \(\sum _j(U_g)_{ij}C^j=VC^iV^\dagger \) obey \(\| \beta _V-\beta _{V_g^0}\| {\lt}\varepsilon \). No continuity or unitarity of the virtual matrices is assumed.

For every \(D{\gt}0\) there is \(\delta _D{\gt}0\) such that the following holds for any group \(G\). Let \(\rho _0,\rho _1\) be unitary projective representations on \(\mathbb {C}^D\), with factors \(\omega _0,\omega _1\). If \(\| \beta _{\rho _1(g)}-\beta _{\rho _0(g)}\| {\lt}\delta _D\) for every \(g\), then \(\omega _1\) is cohomologous to \(\omega _0\). No topology on \(G\) or continuity of either representation is required.

Let \(A_n\) be unital tensors of fixed physical dimension \(d\) and bond dimension \(D\). Suppose that \(\sigma _n\geq 0\), \(\operatorname{tr}\sigma _n=1\), and \(\mathcal E_{A_n}^*(\sigma _n)=\sigma _n\), where \(\mathcal E_A^*(X)=\sum _i(A^i)^\dagger X A^i\). There exist a strictly increasing sequence of indices \(n_k\), a unital tensor \(B\), and a density matrix \(\tau \) such that \(A_{n_k}\to B\), \(\sigma _{n_k}\to \tau \), and \(\mathcal E_B^*(\tau )=\tau \). The rank of \(\tau \) may be smaller than the ranks of the matrices \(\sigma _n\).

Let \(\rho :G\to \mathrm{GL}(D,\mathbb {C})\) be a projective representation with factor \(\omega \), where \(D{\gt}0\). If every \(\rho (g)\) is unitary, then \(\omega (g,h)\overline{\omega (g,h)}=1\) for every \(g,h\).

Let \(A,B\) have the same positive bond dimension, be unital, and be gauge equivalent up to a nonzero scalar, with \(A\) injective. Suppose a unitary physical group representation \(U\) has a unitary projective virtual action \(V\) on \(A\), with factor \(\omega \), satisfying \(\sum _jU(g)_{ij}A^j=V(g^{-1})A^iV(g^{-1})^\dagger \). Then \(B\) admits a unitary projective virtual action with exactly the factor \(\omega \) and the same covariance identity. Source context: [ 236 , Appendix C, lines 2653–2717 ] .

No normal tensor \(A\) on a spin-\(\tfrac 12\) site with \(D\ge 1\) is invariant up to phases under the defining representation of \(\mathrm{SU}(2)\): it is not the case that for every \(g\) and every \(N\) the vector \(U(g)^{\otimes N}|\psi _N(A)\rangle \) is a multiple of \(|\psi _N(A)\rangle \) [ 17 , Section III.A ] . In particular \(A\) has no MPV-family invariance under \(\mathrm{SU}(2)\).

Let \(M\) have columns indexed by pairs \((a,b)\). The associated bilinear map satisfies \(\mu (x,y)=M(x\otimes y)\), where \((x\otimes y)_{(a,b)}=x_a y_b\).

Let \(V^0,V^1\) be unitary projective representations with a common factor system, and put \(V_g=V_g^0\oplus V_g^1\). For the bond coefficient matrices \(W(\gamma )\) and \(\widehat W(\gamma )\) of Theorem 10.4.9.10,

These are the coefficient-matrix identities expressing invariance of the bond vector under \(V_g\otimes \overline{V_g}\) [ 236 , Section II.F.2 ] .

For \(D_0,D_1{\gt}0\) and each \(N{\gt}0\), the independent-bond parent Hamiltonian \(H_N(\gamma )\) is continuous in \(\gamma \in \mathbb R\).

Let \(D_0,D_1{\gt}0\), \(N{\gt}0\), and \(\widehat\omega (\gamma )\) be the normalized bond vector of Theorem 10.4.9.10. In the Hilbert space of \(N\) independent two-leg bonds, let

The terms \(h_i(\gamma )\) are commuting orthogonal projections. Their common kernel is precisely the line spanned by the product vector \(\widehat\omega (\gamma )^{\otimes N}\), and for every vector \(v\),

Thus every nonzero eigenvalue is at least one, uniformly in \(N\) and \(\gamma \) [ 236 , Sections II.D.2 and II.F.2 ] . The same holds for every unit bond vector in place of \(\widehat\omega (\gamma )\). This statement uses independent-bond coordinates; the passage to the original site coordinates uses the unitary of Theorem 10.4.9.13.

For matrices \(L,R\) on a register of dimension \(D\),

In particular, for the physical action \(U(g)=W_g^{\mathsf T}\otimes W_g^{-1}\) of the SPT fixed-point tensor with trivial character (Definition 10.4.7.2), the incoming-bond action is \(W_g^{-1}\otimes W_g^{\mathsf T}\) [ 236 , Section II.F.2 ] .

Let \(\gamma \mapsto A_\gamma \) be a continuous path of one-site injective tensors on fixed physical and bond spaces, with \(D{\gt}0\). Fix a unitary on-site representation \(U\) such that, for every \(\gamma \in [0,1]\) and every \(g\in G\), physical rotation of \(A_\gamma \) by \(U_g\) is a virtual gauge change. Then the canonical two-site parent interactions give a symmetric gapped interaction path between the endpoint interactions. Their norms are at most one, their ground energy is zero, and one positive spectral gap works for every parameter and every ring length \(N\geq 2\) [ 236 , Section II.C and Appendix A ] . The one-site injective restriction is documented in [ 176 ] .

A continuous one-site injective tensor family of fixed positive bond dimension certifies the exact MPS ground lines of any physical path whose local interactions are its canonical two-site parents. At every periodic length at least two the parent kernel is precisely the MPS line, with least energy zero. The periodic vectors are also nonzero at length one. No additional virtual data are required by this ground-state certificate.

Let \(B\) be one-site injective with positive bond dimension and suppose \(\psi _N(A)=c_N\psi _N(B)\) for every \(N\ge 2\), with \(|c_N|=1\). Then \(c_N=(c_3/c_2)^N\) for every \(N\ge 2\). If the coefficients arise from group characters \(c_N(g)\), the ratio \(\chi (g)=c_3(g)/c_2(g)\) is a unit-modulus character, and \(c_N(g)=\chi (g)^N\). When \(A(g)\) and \(B\) have the same bond dimension, the tensors \(\chi (g)B\) and \(A(g)\) have equal periodic vectors at all lengths, including length one. This supplies the one-site phase gauge in [ 236 , Appendix B ] from rings of length at least two.

Multiplying the physical representation by a unit-modulus character preserves the same symmetric gapped interaction path and the same exact MPS ground realization. Both the Hamiltonians and their ground spaces are unchanged. This is the physical phase-gauge freedom of [ 236 , Section II.C.2 and Appendix B ] .

For \(p=0,1\), let \(A_p\) be an injective MPS tensor whose parent Hamiltonian \(H_p\) has a unique ground state and is symmetric under a unitary representation \(U^p\) of \(G\) on \(\mathcal{H}_p\). Choose the endpoint phase gauges. After a common finite blocking, the embedded endpoints admit a continuous symmetric MPS path with corresponding translationally invariant, finite-range parent Hamiltonians that vary smoothly and remain uniformly gapped if and only if their virtual projective representations determine the same class in \(H^2(G,\mathrm U(1))\).

More precisely, for the forward construction there exist a path-sector Hilbert space \(\mathcal{H}_{\mathrm{path}}\) and a linear representation \(U^{\mathrm{path}}\) such that the endpoints embed into

The converse quantifies over smooth paths of MPS with corresponding parent Hamiltonians [ 236 , Section II.F, lines 929–953 ] .

Let \(D_0,D_1{\gt}0\), and let \(V^0\) and \(V^1\) be unitary projective representations with cohomologous factor systems \(\omega _0\) and \(\omega _1\). There is a unitary projective representation \(V^{0\prime }\) with factor system \(\omega _1\) whose fixed-point physical action, with trivial character, equals that of \(V^0\). Set \(D=D_0+D_1\) and \(W_g=V^{0\prime }_{g^{-1}}\oplus V^1_{g^{-1}}\). Under the common on-site action \(U_g=W_g^T\otimes W_g^{-1}\), the normalized bond interactions of Theorem 10.4.9.23 give a symmetric gapped path between their two endpoint interactions on the physical space of dimension \(D^2\) [ 236 , Section II.F.2 ] . The normalized bond follows the corrected second-block range of [ 101 ] .

The symmetric gapped path between the canonical parents of the two included fixed-point tensors has an independent exact MPS ground realization. Its initial and final tensors are precisely the included weighted endpoint tensors. The affine parent comparisons use the constant ground tensor at the corresponding endpoint. More generally, an ordered affine comparison between positive Hamiltonians with the same zero modes preserves that ground line. This certifies the parent comparisons in [ 236 , Section II.F.2 ] .

The prescribed common physical embeddings of the two original tensor spaces have orthogonal ranges. This holds for arbitrary original tensors, without injectivity or symmetry assumptions, and supplies the orthogonal endpoint summands required by [ 236 , Section II.C.2 ] .

The maps \(E_0,E_1,Q\) are isometries. Assume the virtual matrices are unitary and each \(A_j\) has covariance under its original on-site action \(U_j\) and the corresponding virtual action. Then the common action intertwines \(E_j\) with \(U_j\) and \(Q\) with \(V\).

For positive endpoint bond dimensions, the included weighted matrix-unit interpolation is a continuous exact MPS realization of the common-space normalized bond path. Its periodic vectors are nonzero at every positive length. At each interior parameter the tensor is one-site injective on the full direct-sum bond space; at the endpoints it has the corresponding smaller injective representative with the same positive-length periodic rays. Its finite-chain ground eigenspaces are precisely those periodic MPS lines. The ground-state certificate itself requires no virtual representation; symmetry is supplied by the physical path. This verifies the exact MPS interpolation in [ 236 , Section II.F.2 ] in the stated setting.

Let two positive-dimensional unitary virtual representations have the same factor system. In the common physical representation consisting of their direct-sum adjoint action and the two original physical representations, the normalized bond interpolation gives a continuous symmetric interaction path. Each periodic Hamiltonian, at every length \(N\ge 2\), has ground energy zero, gap at least one, and ground space equal to the periodic MPS line of the included weighted matrix-unit tensor. The unused physical states have energy one. Symmetric affine comparisons join its endpoints to the canonical two-site parents of the two included fixed-point tensors. This realizes the common-space construction in [ 236 , Section II.F.2 ] for the stated two-site setting.

Let two one-site injective tensors of positive bond dimension have unitary physical representations and unitary virtual projective representations giving their adjoint tensor covariances. If the actual virtual factor systems are cohomologous, then the canonical two-site parents of their prescribed common physical images are joined by a symmetric uniformly gapped interaction path. The embeddings retain both entire original physical spaces. No chosen cochain or additional target intertwiner is an assumption. This is the one-site injective, exactly covariant construction from [ 236 , Section II.F.2 ] . The independent exact MPS ground realization is constructed in Definition 10.4.1.29.

For \(j=0,1\), the frame identity is \(E_jF_j=QJ_j\). The embedded isometric polar tensor, multiplied by \(D_j^{-1/2}\), is therefore the normalized matrix-unit tensor in the frame \(QJ_j\). Their canonical parent interactions agree at every length.

At parameter zero, each polar ground family is exactly the included weighted fixed-point tensor. At parameter one, its positive-length periodic vector lines are those of the corresponding included original tensor. Thus both fixed-point joins are exact tensor identities, while the original endpoints have the required vector lines.

Suppose symmetric gapped paths connect \(h_0\) to \(h_1\) and \(h_1\) to \(h_2\) on the same physical space with the same on-site action. There is then a symmetric gapped path from \(h_0\) to \(h_2\). If the original gap bounds are \(\delta _0,\delta _1{\gt}0\), the concatenated path has gap bound \(\min \{ \delta _0,\delta _1\} \).

If \(h_\gamma \) is a symmetric gapped path from \(h_0\) to \(h_1\), then \(h_{1-\gamma }\) is such a path from \(h_1\) to \(h_0\), with the same gap bound.

Let \(A\) have physical dimension \(d\) and bond dimension \(D{\gt}0\). Suppose that \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\) and \(\dim \ker (\mathcal{E}_A-\mathbb {1})=1\). Let \(q\in \mathbb {R}\) satisfy \(|\lambda |\leq q\) for every transfer eigenvalue \(\lambda \ne 1\). Then every integer \(N\geq 0\) satisfies

Under the hypotheses of Theorem 10.4.3.7, every integer \(N\geq 0\) satisfies \(1-(D^2-1)q^N\leq \| |V^{(N)}(A)\rangle \| ^2\).

Let \(A\) have dimensions \(d\) and \(D{\gt}0\), with \(\mathcal{E}_A\) unital and \(\dim \ker (\mathcal{E}_A-\mathbb {1})=1\). Let \(P\) be a nonzero orthogonal projection such that \(P A^i(\mathbb {1}-P)=0\) for each physical index \(i\). Put \(R^i=(\mathbb {1}-P)A^i(\mathbb {1}-P)\). Suppose that \(q\geq 0\) and that every eigenvalue of \(\mathcal{E}_A\) other than \(1\) has modulus at most \(q\). Then every eigenvalue of \(\mathcal{E}_R\) has modulus at most \(q\).

Under the hypotheses of Theorem 10.4.3.11, suppose also that \(q{\lt}1\). Then \(\| |V^{(N)}(R)\rangle \| \longrightarrow 0\) as \(N\longrightarrow \infty \).

Under the hypotheses of Theorem 10.4.3.9, there exists an integer \(L{\gt}0\) such that every \(t\in T\) and every integer \(N{\gt}0\) satisfy \(\| |V^{(N)}(A_t^{[L]})\rangle \| ^2\geq \tfrac 12\) and \(|V^{(N)}(A_t^{[L]})\rangle \ne 0\).

Let \(\{ A_t:t\in T\} \) be an arbitrary family of tensors with common dimensions \(d\) and \(D{\gt}0\). Suppose that every \(\mathcal{E}_{A_t}\) is unital, has a one-dimensional fixed space, and has all eigenvalues other than \(1\) of modulus at most a common \(q\) with \(0\leq q{\lt}1\). There exists an integer \(L{\gt}0\) such that \(\| |V^{(N)}(A_t)\rangle \| ^2\geq \tfrac 12\) for every \(t\in T\) and every integer \(N\geq L\).

Under the preceding hypotheses, the neighborhood may be chosen so that the unital tensor \(B(t)\) also has a continuous normalized adjoint stationary density \(\sigma (t)\) satisfying

This density is unique among all trace-one fixed matrices. The one-dimensional fixed-space property is derived, rather than assumed. These are local canonical data for the supplied tensor family; reconstruction from a physical uniform gap remains separate [ 101 ] .

Under the preceding hypotheses, the local representatives can be chosen continuous, one-site injective, and unital, with a continuous normalized adjoint stationary density \(\sigma (t)\). This density is positive semidefinite, has trace one, and is unique among all trace-one fixed matrices of the adjoint transfer map. The minimal bond dimension is constant and positive. Reconstruction through a change of minimal dimension remains separate [ 101 ] .

A continuous matrix family of constant rank \(D\) admits a local continuous frame \(C(t)=P(t)F\) for its range, with a fixed matrix \(F\). The matrices \(H(t)=(C(t)^\dagger C(t))^{-1}C(t)^\dagger \) are continuous and satisfy \(H(t)C(t)=I_D\). For an idempotent family it suffices that \(P(t_0)\) have rank \(D\); in addition \(P(t)C(t)=C(t)\).

Suppose unitary projective matrices, continuous in the parameter for each group element, act on one fixed ambient space, and commutes with a continuous fixed-rank family of orthogonal projections of nonzero rank. Near a prescribed parameter, continuous isometric frames give continuous unitary compressed projective matrices with exactly the original factor system. A family of invertible projective matrices, continuous in the parameter for each group element, on a positive fixed dimension also has continuous scalar factors. The continuous ambient representation and projections are explicit auxiliary hypotheses for the argument in [ 236 , Appendix C ] .

Let \(P\) be the physical matrix of a one-site injective tensor \(A\). Then \(P^\dagger P\) is positive definite, and

Thus \(C_A\) expands every virtual matrix in the tensor letters. It depends continuously on a continuous injective family. Replacing the letters of \(A\) by gauge-equivalent letters of \(B\) in this expansion defines their virtual conjugation action, also continuously. This supplies the inverse of the injective tensor map used in [ 236 , Section II.F.2 ] .

Continuous multiplication coefficients, frames, left inverses, and letter coordinates give continuous represented letters. The letter coordinates are continuous wherever the fixed section is injective.

Let \(B(t)\) be continuous in a fixed ambient bond dimension. Suppose pointwise one-site injective tensors \(A(t)\) have a fixed positive bond dimension \(D\) and satisfy

with nonzero factors. Around each parameter there is a continuous one-site injective tensor \(C(t)\) of dimension \(D\) that is gauge-equivalent to \((\alpha _3(t)/\alpha _2(t))A(t)\). The pointwise tensors, scalar factors, and gauges need not be continuous. This is a reconstruction at constant minimal dimension; the two trace identities alone make no assertion about the raw tensor’s longer periodic rays.

Suppose the continuous raw family \(B(t)\) has, at every parameter, the same positive-length periodic rays as a pointwise one-site injective tensor \(A(t)\) of a fixed positive bond dimension. Locally those same rays admit continuous one-site injective representatives of that bond dimension. No continuity of the supplied pointwise representatives is assumed.

Let \(X(t)\) be a continuous family of complex square matrices, with \(X(t_0)\) idempotent. On an open neighborhood of \(t_0\) there is a continuous family of idempotents \(P(t)\), with \(P(t_0)=X(t_0)\), belonging pointwise to the unital algebra generated by \(X(t)\). No Hermiticity assumption is required.

Let \(P(t)\) be a continuous family of orthogonal projections of rank \(r\), and let \(J_0\) be an isometric frame with \(J_0J_0^\dagger =P(t_0)\). Near \(t_0\), the polar isometry \(J(t)\) of \(P(t)J_0\) depends continuously on \(t\) and satisfies \(J(t)^\dagger J(t)=I\) and \(J(t)J(t)^\dagger =P(t)\). The positive polar factor is continuous for every continuous rectangular matrix family; its polar isometry is continuous wherever the matrices are injective. These are auxiliary ingredients for the support argument in [ 236 , Appendix C ] . Continuity and fixed rank of \(P\) are hypotheses here; deriving the relevant canonical supports from the physical gap remains open [ 101 ] .

Let \(A(t)\) be a continuous family of one-site injective tensors in a fixed positive bond dimension. Its positive right Perron matrix \(\rho (t)\) and positive Perron value \(r(t)\) admit continuous choices satisfying

In fact, any pointwise choice of positive trace-one Perron matrices is continuous. This is a consequence of positive-map Perron–Frobenius theory, applied to the supplied tensor family [ 244 , Theorem 6.3 ] .

Let \(A(t)\) be a continuous Hermitian matrix family and let \(a{\lt}b\). Suppose every spectrum is contained in \((-\infty ,a]\cup [b,\infty )\). A continuous scalar function equal to zero below \(a\) and one above \(b\) gives, by functional calculus, the orthogonal spectral projection \(P(t)\) onto the part above the cutoff. This projection is continuous and has locally constant rank. Every operator commuting with \(A(t)\) commutes with \(P(t)\); conjugation by a commuting unitary fixes the corner projection. The full density rank may change below the cutoff. These facts provide auxiliary separated supports for the argument in [ 236 , Appendix C ] ; continuity and spectral separation of the relevant density remain explicit hypotheses.

Suppose a continuous raw tensor family has two-site and three-site traces proportional, with nonzero factors, to those of pointwise injective tensors of constant positive bond dimension \(D\). Its locally reconstructed trace-quotient algebra admits a continuous idempotent \(p(t)\) whose right multiplication has rank \(D\). Both the multiplication coordinates and this right-multiplication matrix are continuous. No continuous choice of the pointwise minimal tensors or algebra identifications is assumed.

Let \(A(t)\) be continuous in a fixed positive bond dimension and one-site injective at \(t_0\). On an open neighborhood of \(t_0\), there are continuous Perron data as above and a continuous tensor

The tensor \(B(t)\) is unital and is gauge-equivalent to \(r(t)^{-1/2}A(t)\). The continuous tensor family is a hypothesis; no choice from physical ground vectors is asserted here.

Let \(D_0,D_1{\gt}0\), and let \(V^{(0)}\) and \(V^{(1)}\) be unitary projective representations with the same factor system. Set \(D=D_0+D_1\) and let \(U\) be the unitary physical action of the SPT fixed-point tensor of \(V^{(0)}\oplus V^{(1)}\) with trivial character. If \(\widehat\omega (\gamma )\) is the normalized interpolating bond, define the two-site interaction to be the identity on the outer registers and \(1-|\widehat\omega (\gamma )\rangle \langle \widehat\omega (\gamma )|\) on the bond joining the sites. Then these interactions form a symmetric gapped path on the common physical space of dimension \(D^2\). Their operator norms are at most one, and the gap above the unique ground line is at least one for every \(N\ge 2\) and every \(\gamma \in [0,1]\) [ 236 , Section II.F.2 ] .

Let \(h_t\) be the normalized independent-bond interaction and let \(B\) have its physical alphabet. If \(h_t\) annihilates \(G_2(B)\), then \(h_t\leq p_B\), where \(p_B\) is the canonical two-site parent projection. In particular, this holds when an isometric bond inclusion intertwines \(A_t\) with \(B\). If the periodic vectors of \(B\) and \(A_t\) agree at every length \(N\geq 2\), every periodic zero mode of \(h_t\) is annihilated by the canonical parent of \(B\). Under the common unitary physical symmetry of two unitary projective representations with the same factor system and trivial character, covariance of \(B\) therefore makes \(s\mapsto h_t+s(p_B-h_t)\) a symmetric gapped path with gap at least one [ 236 , Section II.F.2 ] .

Fix a linear physical map \(E:\mathbb {C}^d\to \mathbb {C}^m\). The map \(A\mapsto EA\) on tensors is continuous. Suppose physical operators \(U\) and \(V\) obey \(VE=EU\), and an invertible bond matrix \(X\) satisfies \((UA)^i=XA^iX^{-1}\). Then \((V(EA))^i=X(EA)^iX^{-1}\). This transports the covariance into the common physical space of [ 236 , Section II.F.2 ] .

Let \(A\) be injective with \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), let \(\Lambda \) be a positive definite matrix with \(\mathcal{E}_A^\dagger (\Lambda )=\Lambda \), and let \(U\) be a unitary representation of \(G\) under which \(A\) is symmetric with virtual projective representation \(\rho \) up to nonzero scalars \(\zeta (g)\), that is, \(\sum _jU(g)_{ij}A^j=\zeta (g)\, \rho (g^{-1})A^i\rho (g^{-1})^{-1}\). If the factor system of \(\rho \) has a non-trivial class, no eigenspace of \(\Lambda \) is one-dimensional: every eigenvalue of the entanglement spectrum is degenerate [ 17 , Section III.A ] .

In the setting of Theorem 10.4.10.2, let \(u,v\) be unitaries with virtual gauges \(X,Y\), that is, \(\sum _j u_{ij}A^j=\zeta \, XA^iX^{-1}\) and \(\sum _j v_{ij}A^j=\eta \, YA^iY^{-1}\) with \(\zeta ,\eta \neq 0\). If \(XY=-YX\), then every eigenspace of \(\Lambda \) has even dimension. This sharpens Theorem 10.4.10.4 when the projective representation contains an anticommuting pair.

Concatenation retains the common ambient bond dimension. The initial tensor of the three-stage realization is the final tensor of the first, unreversed family, and its final tensor is the final tensor of the third family.

Equality of positive-length MPS vector lines is reflexive, symmetric, and transitive. Reversing an interaction path and its exact MPS realization preserves all its ground-line conditions. In particular, the independent exact MPS phase condition is symmetric.

Let \(A_0,A_1\) be one-site injective tensors of positive bond dimension with unitary physical actions. Suppose actual invertible virtual projective representations implement their exact tensor symmetries. If their factor systems are cohomologous, the tensors satisfy the independent exact MPS phase condition. The whole endpoint physical spaces have orthogonal isometric images in the constructed common physical representation. No virtual family or physical path is assumed. This proves the forward construction of [ 236 , Section II.F.2 ] in the stated one-site, trivial-character setting.

Suppose that \(B_j\) is related to \(c_j A_j\) by an invertible bond change of basis, where \(c_j\ne 0\), for \(j=0,1\). If \(B_0\) and \(B_1\) satisfy the exact MPS phase condition for the endpoint physical representations, then so do \(A_0\) and \(A_1\). The same blocking, characters, orthogonal physical inclusions, interaction path, and continuous ground-state family give the conclusion.

Suppose the full endpoint physical representations have orthogonal isometric inclusions in a common physical representation, intertwining the on-site actions. A symmetric gapped path between their included canonical parents, with an exact MPS ground realization whose endpoint vector lines are the included endpoint lines, proves the independent exact MPS phase condition with blocking length one and trivial endpoint characters.

Let an injective tensor of positive bond dimension have an actual invertible virtual projective implementation of its exact physical symmetry. A nonzero scalar rescaling and a bond change of basis give an injective tensor with a unitary virtual implementation in the same cohomology class. Its physical covariance is adjoint conjugation.

Suppose two one-site injective tensors have proportional periodic vectors at lengths two and three, with nonzero constants \(c_2,c_3\). They have the same minimal bond dimension, and \(B\) is related by a bond conjugation to \((c_3/c_2)A\). Consequently their periodic vector rays agree at every positive length. This finite-data assertion does not construct continuous canonical coordinates through a changing bond dimension.

Let \(A,B\) be one-site injective and suppose their two-site traces satisfy the preceding proportionality, allowing \(c_2=0\) for the purely linear assertion. There is a unique linear map \(T:M_D(\mathbb {C})\to M_E(\mathbb {C})\) with \(T(A^i)=B^i\). If \(c_3\ne 0\) and \(\operatorname{tr}(B^iB^jB^k)=c_3\operatorname{tr}(A^iA^jA^k)\) for all \(i,j,k\), then \(T(MN)=(c_2/c_3)T(M)T(N)\).

Let \(A^i\in M_D(\mathbb {C})\) and \(B^i\in M_E(\mathbb {C})\) be one-site injective. Suppose \(c_2\ne 0\) and \(\operatorname{tr}(B^iB^j)=c_2\operatorname{tr}(A^iA^j)\) for all physical indices \(i,j\). Then the coefficient maps \(x\mapsto \sum _i x_iA^i\) and \(x\mapsto \sum _i x_iB^i\) have the same kernel, and \(D=E\). Their trace forms are the compositions of these coefficient maps with the corresponding nondegenerate trace pairings.

Suppose an independent exact MPS ground-state path has a continuous ambient tensor realization that is one-site injective at every parameter. Then one fixed unit-modulus physical character makes the entire family exactly symmetric. Continuous invertible virtual projective matrices and continuous factor systems follow, and the endpoint factor systems are cohomologous. The endpoint classes agree for any representatives of that common rephased symmetry. Neither exact tensor symmetry nor a virtual path is an assumption. The fixed injective realization is the scope restriction; the varying-support converse remains open [ 101 ] .

The map \(g\mapsto U(g)\) is a representation of \(G\). For every \(g\) and \(i\),

Thus the tensor is on-site symmetric up to \(\phi \), with the vector transformation (71). For \(D\geq 1\), there is a virtual projective representation realizing (72), and every virtual projective representation realizing that relation with the same character \(\phi \) has a factor system cohomologous to \(\omega \). If \(\rho \) is unitary and \(|\phi (g)|=1\) for every \(g\), then \(U\) is a unitary representation [ 17 , Section III.A ] .

Let \(V^0,V^1\) be projective representations with the same factor system on spaces of positive dimensions \(D_0,D_1\), and put \(V=V^0\oplus V^1\). The coordinate inclusions \(J_0,J_1\) of their virtual-pair spaces into \(\mathbb {C}^{(D_0+D_1)^2}\) intertwine the corresponding pair actions with the common fixed-point physical action induced by \(V\). The scalar character is taken to be trivial, as recorded in [ 186 ] . This is the coordinate form of the joint symmetry in [ 236 , Section II.F.2 ] .

Let \(G\) be a finite group, \(\phi :G\to \mathbb {C}^\times \) a character, and \(\omega \) a 2-cocycle. There is a unitary projective representation \(\rho \) of \(G\) on \(\mathbb {C}^G\) whose factor system is cohomologous to \(\omega \). With \(D=|G|\), the physical action \(U\) of Definition 10.4.7.2 is unitary, the SPT fixed-point tensor is on-site symmetric up to \(\phi \), and every virtual projective representation realizing the symmetry with this character has a factor system cohomologous to \(\omega \) [ 17 , Section III.A ] .

Let \(D\geq 1\). The transfer map of the SPT fixed-point tensor is \(X\mapsto D^{-1}\operatorname{tr}(X)\, \mathbb {1}\), an idempotent of rank one, and the tensor is injective, hence normal, so its letters are linearly independent [ 17 , Section III.A ] .

Let \(D{\gt}0\), let \(A^{(a,b)}=D^{-1/2}|a\rangle \langle b|\) be the SPT fixed-point tensor, and set \(\eta =D^{-1/2}\sum _{a=1}^D|a,a\rangle \). Then \(\| \eta \| =1\). For every boundary matrix \(X\), the two-site open-boundary vector \(\Psi _X\) has coefficients \(\Psi _X(a,b,c,d)=D^{-1}\delta _{bc}X_{da}\) and is annihilated by the interaction \(h=I\otimes (I-|\eta \rangle \langle \eta |)\otimes I\) acting on the middle registers. Its kernel is exactly \(G_2(A)\), so \(h\) equals the canonical two-site parent interaction of \(A\) [ 236 , Section II.D.2 ] .

The map \(E\) in (67) satisfies \(E^\dagger E=I\) and \(EF=(J,0)\). If unitary actions obey \(U_gF=FS_g\) and \(V_gJ=JS_g\), then \((V_g\oplus U_g)E=EU_g\). Direct sum preserves multiplication, so two on-site representations give an on-site representation on \(K\oplus H\). For injective \(A\), its isometric polar factor obeys the same covariance as \(A\). Consequently, the physically embedded isometric tensor has exactly the frame \((J,0)\).

Let \(A_t,B_t\), \(t\in [0,1]\), be continuous, pointwise gauge-equivalent tensor families of the same positive bond dimension, with \(A_t\) one-site injective. Given an invertible initial gauge \(X_0\), there is a continuous family of invertible matrices \(X_t\) with that initial value and \(B_t^iX_t=X_tA_t^i\) for every \(t,i\). This proves the interval lifting of the local construction in [ 236 , Section II.F.2 ] .

If a complex rectangular matrix \(C\) has independent columns, then \(H=(C^\dagger C)^{-1}C^\dagger \) satisfies \(HC=I\). For a continuous family with independent columns, \(H\) is continuous.

Let \(P_t\) be a continuous family of rank-one idempotents on a finite-dimensional complex space, with parameter in a connected space. Suppose each \(P_t\) commutes with a fixed unitary representation \(U\) of a group \(G\). There is one unit-modulus character \(\chi \) such that \(U_gP_t=\chi (g)P_t\) for every \(g,t\). No finiteness assumption on \(G\) is needed. Orthogonal projections onto unique ground-state lines are a special case, provided their continuity has been established. This is an ingredient for the phase gauges allowed in [ 236 , Section II.C.2 and Appendix B ] .

An idempotent complex matrix has trace equal to its rank. Consequently the rank of a continuous family of idempotents is locally constant, including for non-Hermitian idempotents.

Let \(N{\gt}0\) and let \(K_i\) act as a matrix \(K\) on incoming bond \(i\) and as the identity on every other bond. Then \(P_NK_iP_N^\dagger \) acts on the registers \((R_{i-1},L_i)\) and as the identity on all other registers. More precisely, its matrix entries between physical configurations \(s,t\) are

Hence every one-bond interaction becomes a nearest-neighbor physical interaction (a one-site interaction when \(N=1\)).

The matrix \(P_N\) defined by \((P_N)_{s,b}=\delta _{e_N(s),b}\) is unitary. Thus it takes a bond-coordinate wavefunction to the corresponding physical-coordinate wavefunction by composition with \(e_N\).

If a two-site interaction commutes with the tensor square of an on-site matrix, then its translation-invariant periodic Hamiltonian commutes with the full tensor power at every chain length at least two. Unitarity is not needed for this algebraic identity.

Let \(V_g\) be a unitary projective representation on \(\mathbb {C}^D\) with factor system \(\omega \), and let \(J:\mathbb {C}^r\to \mathbb {C}^D\) be an isometry, where \(r{\gt}0\). Suppose \(V_g\) commutes with \(JJ^\dagger \) for every \(g\). Then \(J^\dagger V_gJ\) is a unitary projective representation with exactly the same factor system \(\omega \). This is the algebraic restriction in [ 236 , Appendix C ] . The invariant bond subspace is supplied here; deriving it continuously from the physical gap remains a separate problem, recorded in [ 101 ] .

Let \(A,B\) have the same physical alphabet and suppose \(V^\dagger V=I\) and \(A^iV=VB^i\) for every letter. Then \(\Gamma _L^A(VXV^\dagger )=\Gamma _L^B(X)\) for every \(L\), and \(G_L(B)\subseteq G_L(A)\). If also \(VV^\dagger A^i=A^i\), their periodic vectors agree at every positive length. Finally, if invertible matrices \(X,Y\) satisfy \(XV=VY\) and \(\sum _jU_{ij}A^j=XA^iX^{-1}\), then \(\sum _jU_{ij}B^j=YB^iY^{-1}\) [ 236 , Section II.F.2 ] .

Let \(T:H\to K\) be an isometry between finite-dimensional Hilbert spaces, let \(G\subseteq H\) be a subspace, and denote its orthogonal projection by \(P_G\). Then \(P_{TG}=TP_GT^\dagger \).

Let \(h\) be an orthogonal projection and \(E\) a one-site isometry. Its extension \(h'\) is an orthogonal projection. Suppose the periodic translates of \(h\) commute, and those of \(h'\) also commute, on a ring of length \(N\ge 2\). Then, with \(E_N=E^{\otimes N}\),

In particular, a prescribed ground space \(\operatorname {span}\{ \psi \} \) is transported to \(\operatorname {span}\{ E_N\psi \} \). The unused-state penalty introduces no additional zero modes.

Let \(h\) be Hermitian and \(N\ge 2\). With \(E_N=E^{\otimes N}\), the periodic sums of \(h\) and its extension satisfy \(H'_NE_N=E_NH_N\). The same identity holds for each local translate; it also holds for the raw local conjugation \(ThT^\dagger \) on the included chain. Every nonzero zero mode \(\psi \) of \(H_N\) therefore gives a nonzero zero mode \(E_N\psi \) of \(H'_N\), so zero belongs to its spectrum.

Let \(E\) be a physical isometry. The canonical two-site parent of \(EA\) is exactly the isometric interaction extension of the canonical parent of \(A\). Moreover, extension preserves local operator inequalities: \(h\le k\) implies \(h'\le k'\).

Suppose the quotient algebra is isomorphic to \(M_{D}(\mathbb {C})\), with \(D{\gt}0\), and \(J\) identifies \(\mathbb {C}^D\) with a principal left ideal, with \(HJ=I\). The framed action is an algebra isomorphism onto \(M_{D}(\mathbb {C})\) and differs from the given identification by an inner conjugation. It preserves traces. Applying it to a family of letters therefore preserves that family’s bond gauge equivalence.

Under the pointwise compression assumptions, it suffices that \(C(t)\) be continuous on a neighborhood of \(t_0\) and injective at \(t_0\). Then \(\omega _t\) and \(\omega _{t_0}\) are cohomologous for all sufficiently nearby \(t\). For a fixed ambient bond space, continuity of \(A(t)\) and \(J(t)\) on that neighborhood implies the required continuity of \(C(t)=J(t)^\dagger A(t)J(t)\).

Let \(\alpha _t\) be a family of inner actions on \(M_D(\mathbb {C})\), with \(D{\gt}0\), such that \(t\mapsto \alpha _t(M)\) is continuous for every \(M\). Suppose \(\alpha _{t_0}=\operatorname {Ad}X_0\) with \(X_0\) invertible. Fix \(a\in \{ 1,\ldots ,D\} \) and set

Then \(X\) is continuous, \(X(t_0)=X_0\), and \(\alpha _t(M)X(t)=X(t)M\). It is invertible in a neighborhood of \(t_0\). This is an auxiliary local construction for [ 236 , Section II.F.2 ] .

Let \(\sigma (t)\) be a continuous Hermitian matrix family, with \(\sigma (t_0)\geq 0\), and let \(J_0\) be an isometric frame of its support. There are \(0{\lt}a{\lt}b\) and a neighborhood of \(t_0\) on which the spectral projection \(P(t)\) onto the eigenvalues above the cutoffs has constant rank and satisfies \(P(t_0)=\operatorname {supp}\sigma (t_0)\). The polar isometry \(J(t)\) of \(P(t)J_0\) is continuous there, with \(J(t_0)=J_0\), \(J(t)^\dagger J(t)=I\), and \(J(t)J(t)^\dagger =P(t)\). When \(\sigma (t)\geq 0\), the cutoff always satisfies \(\operatorname {supp}\sigma (t)\, P(t)=P(t)\), whether or not its spectrum is separated at that parameter.

Let \(B(t)\) be a continuous family of unital tensors of positive, fixed bond dimension. Suppose the adjoint transfer map at \(t_0\) has a one-dimensional fixed space over \(\mathbb {C}\). Then there is an open neighborhood of \(t_0\) on which it has a unique trace-one stationary matrix \(\sigma (t)\). This matrix is positive semidefinite and continuous; its rank may change. Consequently any pointwise choice of trace-one stationary matrices agrees with this family and is continuous there.

Let \(T\) be a topological space and let \(B(t)\), \(t\in T\), be a continuous tensor family of fixed bond dimension. Let \(\sigma (t)\geq 0\) be trace-one and adjoint stationary. Let the actual minimal tensors be support compressions of \(B(t)\), with unitary projective actions implementing a fixed exact physical symmetry. Suppose the compressed tensor at \(t_0\) is one-site injective. If \(\sigma \) is continuous and unique among trace-one stationary matrices on an open neighborhood of \(t_0\), then the virtual factors at nearby \(t\) are cohomologous to the factor at \(t_0\). Alternatively, it suffices that \(B(t)\) be unital and that its adjoint fixed space at \(t_0\) have complex dimension one. In this case density continuity and local uniqueness follow automatically. Neither case assumes continuity of the minimal support frames or virtual actions.

A continuous pair of pointwise gauge-equivalent one-site injective tensor families of fixed positive bond dimension admits locally continuous invertible bond gauges near any prescribed initial gauge. In particular, if a continuous injective family has exact on-site symmetry, then for each group element its virtual symmetry matrix can be chosen continuously and invertibly near each parameter. No continuous virtual family is assumed. This is the local, fixed-dimension step of [ 236 , Section II.F.2 ] ; interval gluing and changes of minimal bond dimension remain separate steps, as recorded in [ 101 ] .

For a matrix \(K\) acting on the middle registers, a two-site vector \(f\), and exterior indices \(a,d\), one has

In particular, if \(f_{abcd}=\eta _{bc}F_{ad}\), then its image has coefficients \((K\eta )_{bc}F_{ad}\). For a unit vector \(\eta \), \(K=I-|\eta \rangle \langle \eta |\) annihilates every such \(f\) [ 236 , Section II.D.2 ] .

A one-site injective tensor of bond dimension \(D\) has \(\operatorname {rank}C_2(A)=D^2\). Along a continuous exact MPS ground realization, suppose an injective representative at \(t_0\) has positive bond dimension \(D_0\). In a neighborhood of \(t_0\) in the parameter interval, every injective representative of the same positive-length periodic vector lines has bond dimension at least \(D_0\). This proves the local dimension-direction assertion in [ 236 , Section II.F.2, lines 1059–1068 ] . It uses the two-site vector and its continuity, and does not derive a surviving single-bond projective subspace from the physical gap.

Let \(G\) be finite and let \(\phi ,\omega \) be any pair of genuine twisted circle cocycles as above. With \(D=|G|\), the corrected dimer tensor \(A^{(a,b)}=D^{-1/2}|a\rangle \langle b|\) is injective and has a rank-one idempotent transfer map. Let \(P\) swap the two physical dimer factors and define

Then every \(U(g)\) is unitary, \(U(1)=\mathbb {1}\), and \(U(gh)=U(g)C_{t(g)}(U(h))\). The tensor has the mixed symmetry \(S_g(A)^i=\phi (g)X_g^\dagger A^iX_g\). Its periodic \(N\)-site vector transforms by \(\phi (g)^N\), with the actual conjugation and site reversal of (73). Every other unitary virtual action implementing this same symmetry and character has the same class \([\omega ]\in H^2_{t+r}(G,\mathrm U(1))\) [ 17 , Section III.A, local TeX lines 1147–1157 ] .

For every finite group \(G\) and genuine circle-valued two-cocycle \(\omega \), there is a unitary conjugate-projective representation on \(\mathbb C^G\) with factor system exactly \(\omega \). No normalization of the chosen representative is required.

In a complete normed ring with \(\| 1\| \le 1\), set \(x_{n+1}=3x_n^2-2x_n^3\) and \(d_n=x_n^2-x_n\). Then \(d_{n+1}=d_n^2(4d_n-3)\) and \(x_{n+1}-x_n=-(2x_n-1)d_n\). If \(\| d_0\| \le 1/16\), the sequence converges to an idempotent. More precisely,

Let \(D_0,D_1{\gt}0\), and let the two direct-sum bond spaces have these dimensions. Write \(W(\gamma )\) for the coefficient matrix of

For every real \(\gamma \) this matrix is nonzero, and it depends continuously on \(\gamma \). Dividing by \(\sqrt{D_0(1-\gamma )^2+D_1\gamma ^2}\) gives the coefficient matrix \(\widehat W(\gamma )\) of a continuous unit vector. At the two endpoints, only the first and second diagonal blocks, respectively, remain. The second summation range follows the correction recorded in [ 101 ] .

For two unitary projective bond representations with the same factor system, each normalized bond interaction commutes with the two-site physical action of their direct sum. This holds at every interpolation parameter. In addition, embedding an interaction over an entire nonempty ring with origin at the first site leaves it unchanged.

Let \(D{\gt}0\), \(c=D^{-1/2}\), and \(J:\mathbb {C}^{D^2}\to H\) be linear. The normalized matrix-unit tensor embedded by \(J\) has physical matrix \(cJ\). If its physical covariance is induced by virtual multiplication \(B^i\mapsto XB^iY\), then

This identifies the convenient fixed-point basis in [ 236 , Section II.F.2 ] .

Let \(U\) be a unitary on the physical space, and suppose the physical rotation of \(A\) by \(U\) is gauge equivalent to \(A\). The canonical parent interaction of range \(L\) commutes with \(U^{\otimes L}\). For every \(N\geq L\), the periodic parent Hamiltonian commutes with \(U^{\otimes N}\) [ 236 , Section II.F.2 ] .

For a two-site interaction \(h\), let \(H_N(h)\) be the sum of its translates on the periodic chain of length \(N\ge 2\). If \(h\ge 0\), then \(H_N(h)\ge 0\); if \(h\le k\), then \(H_N(h)\le H_N(k)\). For every real \(t\), \(H_N(h+t(k-h))=H_N(h)+t(H_N(k)-H_N(h))\).

Conjugate \(H_N(\gamma )\) by the incoming-bond unitary \(P_N\). In the original physical-site coordinates, the resulting Hamiltonian \(P_NH_N(\gamma )P_N^*\) varies continuously with \(\gamma \), has the transported product vector \(P_N\widehat\omega (\gamma )^{\otimes N}\) as its unique ground line, and satisfies

for every \(N{\gt}0\) and every physical-chain vector \(v\).

For every unit bond vector, and in particular for \(\widehat\omega (\gamma )\) with \(D_0,D_1{\gt}0\), the physical parent Hamiltonian \(P_NH_N(\gamma )P_N^*\) is positive semidefinite for every \(N{\gt}0\). Its spectrum contains \(0\), and every spectral value \(z\) satisfies \(\operatorname {Re}z=0\) or \(\operatorname {Re}z\geq 1\).

Let \(V^{(0)}\) and \(V^{(1)}\) be finite-dimensional unitary projective representations with the same factor system, and let \(V\) be their direct sum. Let \(U\) be the physical action \(U(g)=W_g^{\mathsf T}\otimes W_g^{-1}\), \(W_g=V_{g^{-1}}\), of the SPT fixed-point tensor of \(V\) with trivial character (Definition 10.4.7.2); it does not depend on \(\gamma \). For every group element \(g\) and every \(N{\gt}0\),

[ 236 , Section II.F.2 ] .

Multiplying an on-site unitary representation \(U_g\) by a unit-modulus character \(\chi (g)\) gives another unitary representation. Tensor twisting then acquires the scalar \(\chi (g)\). Consequently, if the twisted periodic vectors of \(A\) equal those of \(\chi (g)A\) at every length, replacing \(U_g\) by \(\chi (g)^{-1}U_g\) makes the symmetry exact at every length. This is the physical phase-gauge change allowed in [ 236 , Section II.C.2 and Appendix B ] .

Let \(E:\mathbb C^d\to \mathbb C^m\) be a linear map and let \((EA)^i=\sum _jE_{ij}A^j\). Write \(T_L=E^{\otimes L}\). Then \(\Gamma _L^{EA}(X)=T_L\Gamma _L^A(X)\) and \(G_L(EA)=T_LG_L(A)\) for every boundary matrix \(X\) and every \(L\ge 0\). If \(E^\dagger E=I\), then \(T_L^\dagger T_L=I\). More generally, adjoints distribute over finite tensor products: \((\bigotimes _j E_j)^\dagger =\bigotimes _j E_j^\dagger \). These identities apply to the physical inclusions in [ 236 , Section II.F.2 ] .

Suppose \(E^\dagger E=I\) and set \(T_L=E^{\otimes L}\). The canonical parent projection satisfies

The identity holds both as an operator and in the configuration basis. In particular, \(T_L^\dagger v=0\) implies \(h_L(EA)v=v\). No injectivity assumption on \(A\) is needed.

If the letters transform by \(A^i\mapsto XA^iY\), their physical matrix transforms by \(M\mapsto M(X^T\otimes Y)\). Thus physical covariance \(\sum _j U_{ij}A^j=XA^iY\) gives \(UM=M(X^T\otimes Y)\). For unitary \(X\) and \(Y=X^\dagger \) this is the bond action \(X^T\otimes X^{-1}\). The physical matrix \(U\) need not be unitary for this algebraic implication.

For an independent exact MPS ground-state path and every \(N\ge 2\), there is one unit-modulus character \(c_N\) such that the physical symmetry acts on every length-\(N\) ground vector by \(c_N(g)\), independently of the path parameter. The ambient tensors need not be injective, and their minimal bond dimensions may change.

Let \(A(t)\) have bond dimension \(D(t)\), and let \(V_g(t)\) be a unitary projective representation with factor \(\omega _t\) implementing exact covariance for a fixed physical action. Let \(J(t):\mathbb {C}^r\to \mathbb {C}^{D(t)}\) be isometric, where \(r{\gt}0\), and suppose \(V_g(t)\) commutes with \(J(t)J(t)^\dagger \). If \(C^i(t)=J(t)^\dagger A^i(t)J(t)\) is continuous and one-site injective for \(0\leq t\leq 1\), then \(\omega _1\) and \(\omega _0\) are cohomologous. No continuity of \(D(t)\), \(A(t)\), \(J(t)\) or \(V_g(t)\) is required, and the tensor letters need not preserve the included bond subspace.

Suppose \(A\) is one-site injective. Then \(P_\gamma =(1-\gamma )I+\gamma P\) is positive definite, \(A_\gamma \) is injective, and the path is continuous, with \(A_0=W\) and \(A_1=A\). If unitaries \(U\) and \(K\) satisfy \(UM=MK\), then \(P\) commutes with \(K\), \(UV=VK\), and \(UM_\gamma =M_\gamma K\). In particular, if \(U\) acts on \(A\) by conjugation with a unitary bond matrix \(X\), it acts on every \(A_\gamma \) by the same conjugation [ 236 , Section II.C, isometric form and symmetries ] .