Dependencies

If \(D {\gt} 0\), the factor system \(\omega \) from Theorem 11.1.16 satisfies the 2-cocycle identity for all \(g,h,k \in G\):

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

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 \(\beta _\omega (g,h):=\omega (g,h) \omega (h,g)^{-1}\). It is defined for any \(\mathbb {C}^{\times }\)-valued \(2\)-cochain; the cocycle condition is not needed.

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

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

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

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

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

The second cohomology set is the quotient

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

A tensor has physical string order in the sense of [ PGWS\(^{+}\)08 , display SOP, lines 112–122 ] if there exist a unitary \(u\ne \mathbb {1}\) and physical endpoint operators \(x,y\) for which Definition 11.3.5 holds. The comparison with arbitrary virtual boundary matrices is a separate endpoint- realizability statement.

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.

Virtual-boundary nondecay is an existence statement over the virtual matrices \(X,Y\), for a fixed twist \(u\); no claim is made for any particular boundary choice. Definition 11.3.6 records the physical-endpoint string-order notion from the source, which quantifies existentially over \(u,x,y\) and requires a positive limit rather than a lower bound uniform in the length. That paper also notes that the correlator for a particular endpoint choice can vanish even when the physical symmetry is present.

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

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

exactly, and \(\omega _A\sim \omega _B\). Thus the predicate is equality of virtual factor-system classes. It neither quantifies over Hamiltonian paths nor states physical phase equivalence.

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.

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

The predicate called on-site symmetry requires \(\mathcal{V}(A)=\mathcal{V}(\widetilde A_g)\) for every \(g\in G\), where \(U:G\to \mathrm{GL}_d(\mathbb {C})\) acts on the physical index. This is equality of periodic MPV families and is the hypothesis used to construct virtual local gauges. It is not, by itself, the source notion that every reduced density operator of an infinite physical state is invariant under \(U(g)^{\otimes N}\).

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 [ PGWS\(^{+}\)08 , 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 \(g,h \in G\),

The relation called projective equivalence holds precisely when the two factor systems are cohomologous. It compares their classes in \(H^2(G,\mathbb {C}^\times )\); it does not assert an intertwining equivalence between the underlying representation spaces.

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

Identifying this periodic overlap with the normalized stationary-boundary functional \(T_L(A,u,\Lambda )\) requires a thermodynamic-boundary and normalization hypothesis. Thus Definition 11.3.2 must not be read as the literal source definition of \(R_L(u)\).

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 11.3.2. The matrices \(X,Y\) act on the virtual level. In the physical string correlator of  [ PGWS\(^{+}\)08 ] 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 [ PGWS\(^{+}\)08 ] 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 \(R_L(u;X,Y) \longrightarrow 0\).

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

For a word \(b:\{ 0,\ldots ,L-1\} \to \{ 0,\ldots ,d-1\} \),

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. 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 \(u\).

This is Lemma 1 of  [ PGWS\(^{+}\)08 ] . 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.

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

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 \((A,B_u)\):

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 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

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 \(V\) 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. In fact, covariance and \(VV^\dagger =\mathbb {1}\) alone imply the conclusion; the injectivity hypothesis is unnecessary.

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 tensor and assume that \(A\) and \(B\) generate the same MPV family. 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 is a nonzero scalar \(u \in \mathbb {C}^{\times }\) such that \(Y=u\cdot X\).

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 \(g\).

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 \(\sigma \) 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 \(A\) 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 11.3.18, 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.

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

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

This is Theorem 1 of  [ PGWS\(^{+}\)08 ] . A pure finitely correlated state has physical string order if and only if there exist a unitary \(\widetilde u\ne \mathbb {1}\), a unitary virtual matrix \(V\), and physical indices \(n,m\) such that

The transfer criteria below assume one-site injectivity. The literal physical equivalence additionally requires that sufficiently long physical endpoint operators realize the needed virtual boundary matrices.

This is Theorem 2 of  [ PGWS\(^{+}\)08 ] . For a pure finitely correlated state and a unitary \(u\ne \mathbb {1}\), the physical state is invariant under \(u^{\otimes N}\) on every \(N\)-site reduced density operator if and only if

The theorem below gives the corresponding virtual local-covariance criterion for a one-site-injective tensor. Passing between that condition and invariance of every physical reduced state is an additional step.

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 \(c\in \mathbb {C}\).

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 whose unique ground states remain MPS and whose Hamiltonians are the corresponding parent Hamiltonians, as in [ SPGC11 , Section II.F, lines 929–953 ] . It does not assert invariance along an arbitrary gapped Hamiltonian path.

For an injective MPS, 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-boundary nondecay for \(u\) holds if and only if \(A,u,\Lambda \) satisfy virtual local covariance.

Let \(A\) be injective, let \(\Lambda \in M_{D}(\mathbb {C})\), and assume \(u u^\dagger = \mathbb {1}\) and \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\). If virtual-boundary nondecay holds for \(u\) with boundary parameter \(\Lambda \), then \(A\) is gauge-phase equivalent to the companion tensor \(B_u\).

Assume \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\) and that, for every physical index \(i\), \(\sum _j u_{ij} A^j = \mu \, V A^i V^\dagger \) with \(V\) unitary. Then \(V\) is an eigenmatrix of the twisted transfer map:

Let \(A\) be an injective MPS tensor with \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), and let \(u\) satisfy \(u u^\dagger = \mathbb {1}\). If

for some nonzero \(X \in M_{D}(\mathbb {C})\) with \(|\lambda |=1\), then \(A\) is gauge-phase equivalent to the companion tensor \(B_u\).

Let \(A\) be an injective MPS tensor with \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), and let \(u\) satisfy \(u u^\dagger = \mathbb {1}\). If \(\mathcal{E}_u(X) = \lambda X\) for some nonzero \(X \in M_{D}(\mathbb {C})\), then \(|\lambda | \le 1\).

Let \(A\) be an injective MPS tensor that is on-site symmetric under a group representation \(U : G \to \mathrm{GL}_{d}(\mathbb {C})\). Then there exist:

1. a scalar 2-cochain \(\omega : G \times G \to \mathbb {C}^{\times }\), and

2. a map \(\rho : G \to \mathrm{GL}_{D}(\mathbb {C})\) satisfying, for all \(g,h \in G\),

   \begin{align} \rho (g) \rho (h) & =\omega (g,h) \rho (gh). \notag \end{align}

such that, defining the gauge matrices \(X(g):=\rho (g^{-1})\), one has, for every \(g \in G\) and \(i \in \{ 0,\ldots ,d{-}1\} \),

In tensor-network notation, for each fixed \(g\) this is the local gauge relation

with the black node denoting the tensor named in the surrounding formula and the two red side nodes acting on the virtual legs. In particular, \(\rho \) is a projective representation of \(G\) on the bond space \(\mathbb {C}^D\).

If \(A\) is injective and on-site symmetric under \(U\), then for every \(g \in G\) there exist an invertible matrix \(X(g) \in \mathrm{GL}_{D}(\mathbb {C})\) and a nonzero scalar \(\varphi (g) \in \mathbb {C}^{\times }\) (in fact \(\varphi =1\) in the single-block case) such that, for every physical index \(i\),

For an injective MPS, let \(\Lambda \in M_{D}(\mathbb {C})\), and assume \(u u^\dagger = \mathbb {1}\) and \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\). If virtual-boundary nondecay holds for \(u\) with boundary parameter \(\Lambda \), then there exists a virtual unitary \(V\) and a unit-modulus scalar \(\mu \) such that \(\sum _j u_{ij} A^j = \mu \, V A^i V^\dagger \) for all \(i\). Diagrammatically this is the same local intertwining relation, but only up to an overall unit scalar on the virtual side:

\(\; =\; \mu \, \) .

Let \(A\) be injective, assume \(u u^\dagger = \mathbb {1}\) and \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\). If \(A\) is gauge-phase equivalent to the companion tensor \(B_u\), then there exist a unitary \(V\) and a phase \(\mu \) with \(|\mu |=1\) such that, for every physical index \(i\),