Dependencies

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 \(g,h \in G\),

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.

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: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

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

The mixed transfer operator (or cross transfer operator) for two MPS tensors \(A\) and \(B\) of the same bond dimension \(D\) is the linear map \(F_{AB}:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) defined by

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

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

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

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

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

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

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.

An action of a monoid \(G\) on the physical index set \(I\) is a map \(\sigma : G \to \mathrm{End}(I)\) with \(\sigma (1) = \operatorname{id}\) and \(\sigma (gh) = \sigma (g) \circ \sigma (h)\) for all \(g, h \in G\). When \(G\) is a group, each \(\sigma (g)\) is a permutation of \(I\), with inverse \(\sigma (g^{-1})\).

Given an action \(\sigma \) of \(G\) on \(I\), the permutation-twisted tensor at the element \(g\) is

In tensor-network notation the local tensor is unchanged and only the physical leg is relabelled:

\(\qquad \xrightarrow {\sigma _g} \qquad \) .

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 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

A common trace-preserving gauge setup for \((A,u)\) consists of a tensor \(B\) equal to the companion \(B_u\) and satisfying \(\mathcal{E}_B(X)=\mathcal{E}_A(X)\) for every \(X\in M_{D}(\mathbb {C})\). It also includes a positive definite matrix \(\sigma \) fixed by the adjoint transfer map of \(B\), its positive square root \(S\), and gauged tensors \(A',B'\) satisfying

In addition,

together with irreducibility of both \(A'\) and \(B'\). These identities complete the common trace-preserving gauge data.

Let \(D{\gt}0\). If \(A\) is injective, \(uu^\dagger =\mathbb {1}_d\), and \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), define the common trace-preserving gauge setup for \((A,u)\) by choosing the unique positive definite stationary state of the adjoint channel and using the data in Definition G.5.2.

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 \(Z = \lambda \mathbb {1}_D\).

A scalar \(2\)-cochain \(\omega \) is a coboundary if and only if \(\omega \sim \mathbb {1}\), where \(\mathbb {1}(g,h)=1\) for all \(g,h\).

Diagrammatically, the hypothesis is that the two local gauges

produce the same target tensor \(B^i\) from the same source tensor \(A^i\). If \(X, Y \in \mathrm{GL}_D(\mathbb {C})\) both conjugate an MPS tensor \(A\) to the same tensor \(B\) (i.e. \(B^i = X A^i X^{-1} = Y A^i Y^{-1}\) for all \(i\)), then \(Y^{-1} X\) commutes with every \(A^i\).

Under the same hypotheses, if \(A\) is additionally injective, then \(Y^{-1} X = \lambda \mathbb {1}_D\) for some scalar \(\lambda \in \mathbb {C}\).

For \(M\in \mathbb {C}^{m\times d}\) and a length-\(N\) physical configuration \(s=(s_0,\ldots ,s_{N-1})\in \{ 0,\ldots ,m-1\} ^N\), physical rotation satisfies

If virtual-boundary nondecay holds for \(A,u,\Lambda \), then for some boundary matrices \(X,Y\) the sequence \(R_L(u;X,Y)\) does not tend to \(0\) as \(L\to \infty \).

For an action \(\sigma \) and all \(g, h \in G\),

For an action \(\sigma \), \((A^{\sigma _1})^i = A^i\) for all \(i\).

If \(uu^\dagger =\mathbb {1}_d\), then for every \(X\in M_{D}(\mathbb {C})\) the companion family \(B_u\) satisfies

The twisted tensor satisfies the composition law

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

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:

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 \(W\):

The relation “cohomologous to” is reflexive, symmetric, and transitive on the set of \(\mathbb {C}^\times \)-valued scalar \(2\)-cochains, and therefore also on the subset of cocycles.

Let \(A\) be an injective MPS tensor with on-site symmetry under a group representation \(U : G \to \mathrm{GL}_d(\mathbb {C})\). For each \(g \in G\), let \(X(g) \in \mathrm{GL}_D(\mathbb {C})\) satisfy \(\widetilde{A}_g^i = X(g)\, A^i\, X(g)^{-1}\) for all \(i\). Then for all \(g, h \in G\), there exists \(c(g,h) \in \mathbb {C}\) such that \(X(h)\, X(g) = c(g,h)\, X(g \cdot h)\). Invertibility of both sides forces \(c(g,h) \in \mathbb {C}^{\times }\). Diagrammatically, both candidate gauges realize the twisted tensor \(\widetilde{A}_{gh}^i\):

\( = \)

and both sides conjugate \(A^i\) to the same twisted tensor \(\widetilde{A}_{gh}^i\).

If \(A\) and \(B\) are injective, both on-site symmetric under a unitary representation \(U\), and if \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), \(\mathcal{E}_B(\mathbb {1})=\mathbb {1}\), \(\Lambda _A\) and \(\Lambda _B\) are positive definite, \(\operatorname{tr}(\Lambda _A)=1\), \(\operatorname{tr}(\Lambda _B)=1\), \(\mathcal{E}_A^\dagger (\Lambda _A)=\Lambda _A\), and \(\mathcal{E}_B^\dagger (\Lambda _B)=\Lambda _B\), then for every \(g \in G\), virtual-boundary nondecay holds for \(A\) with twist \(U(g)\) if and only if it holds for \(B\). In particular, since Definition 10.4.1 implies on-site symmetry for both tensors, this theorem applies to tensors in the same SPT phase; it applies equally to symmetric tensors in different SPT phases.

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

If \(V\ne 0\) and \(\mathcal{E}_u(V)=\lambda V\), then \(\lambda \) is an eigenvalue of the mixed transfer map \(F_{A',B'}\) in any common trace-preserving gauge setup.