12 Symmetries, Physical String Order, and Virtual-Boundary Nondecay

This chapter collects the symmetry-theoretic consequences of the MPS Fundamental Theorem [ PGVWC07 ] , following the symmetry analysis of [ PGWS$^{+}$08 ] . For injective tensors, equality of periodic MPV families under a physical index action determines virtual gauges, and those gauges define projective representations on the bond space. We keep this virtual covariance distinct from symmetry of an infinite physical state, and distinguish physical endpoint string order from virtual-boundary nondecay.

12.1 Virtual Local Covariance and Projective Bond Actions

With a full group of on-site symmetries (rather than just a single unitary), the resulting gauges determine a projective representation on the bond space.

Definition 12.1.1 Twisted tensor

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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

\begin{align} \widetilde{A}_g^i & :=\sum _{j=0}^{d-1}U(g)_{ij}\, A^j. \label{eq:symmetry_twisted_tensor} \end{align}

Definition 12.1.2 MPV-family invariance under physical twists

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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 $U(g)^{\otimes N}$.

Lemma 12.1.3 Functoriality of twisting

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The twisted tensor satisfies the composition law

\begin{align} \widetilde{A}_{gh} & =\widetilde{(\widetilde{A}_h)}_g, \label{eq:symmetry_twist_composition} \end{align}

i.e. twisting by $gh$ is the same as first twisting by $h$ and then by $g$.

Proof ▶

Expand both sides of (??) using (??) and $U(gh)=U(g)\, U(h)$, then swap the order of summation.

Lemma 12.1.4 Identity twist

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

Proof ▶

Since $U(1)=\mathbb {1}_d$, each component in (??) reduces to $\sum _j\delta _{ij}\, A^j=A^i$.

Definition 12.1.5 Blocked on-site action

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

\begin{align} U^{[L]}(g) & :=U(g)^{\otimes L} \in \mathrm{GL}_{d^L}(\mathbb {C}). \notag \end{align}

Lemma 12.1.6 Twisting commutes with blocking

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For every $g\in G$ and $L\ge 0$,

\begin{align} \widetilde{(A^{[L]})}^{\, U^{[L]}}_g & =(\widetilde A_g)^{[L]}. \label{eq:symmetry_twist_block_comm} \end{align}

Proof ▶

Decode a blocked physical index as a word of length $L$ and use the word expansion from Lemma H.1.1.3. Reindexing the sum over words by the blocked index gives exactly the Kronecker-product matrix coefficients on the left of (??).

Theorem 12.1.7 MPV-family symmetry is preserved by blocking

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

Proof ▶

Equation (??) identifies the twisted blocked tensor with the blocking of the twisted tensor. Equality of MPV families is preserved by blocking, so the defining equality for $A$ yields the defining equality for $A^{[L]}$.

Lemma 12.1.8 Symmetric twists are gauge equivalent

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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 $A$.

Proof ▶

On-site symmetry gives $\mathcal{V}(A)=\mathcal{V}(\widetilde{A}_g)$, and the single-block Fundamental Theorem turns this equality of MPV families into gauge equivalence.

Remark 12.1.9 Fundamental-Theorem route for injective symmetry

The injective virtual-representation theorem is not proved from virtual-boundary nondecay. The on-site symmetry condition first identifies $A$ and $\widetilde{A}_g$ as the same MPV family; the single-block Fundamental Theorem gives the gauge in Lemma 12.1.8. Gauge uniqueness and the twist composition law then give Theorem 12.1.16. The virtual-boundary nondecay results later in this chapter use this virtual representation as an input.

Theorem 12.1.10 Virtual symmetry equation

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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 $i$,

\begin{align} \widetilde{A}_g^{\, i} & =\varphi (g)\, X(g)\, A^i\, X(g)^{-1}. \label{eq:symmetry_virtual_equation} \end{align}

Proof ▶

The physical action is transferred to the virtual level by the same local move that replaces the twisted tensor by a gauge-conjugated one:

$\begin{tenkz}[rows={op:none,ket}] \tn[role=operator,up=$i$]{U(g)} \\ \tn{A} \end{tenkz}$ $\qquad \leadsto \qquad $ $\begin{tenkz}[physical=up] \tnX{X(g)} & \tn[up=$i$]{A} & \tnX{X(g)^{-1}} \end{tenkz}$ .

Apply Lemma 12.1.8 to obtain $X(g)$, then set $\varphi (g)=1$ in (??).

Lemma 12.1.11 Gauge uniqueness up to scalar

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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 $Y=u\cdot X$.

Proof ▶

The matrix $Z=Y^{-1}X$ commutes with every $A^i$. Since $A$ is injective, $\{ A^i\} $ spans $M_{D}(\mathbb {C})$, so $Z$ lies in the centre of $M_{D}(\mathbb {C})$. By Lemma 2.1.5, $Z$ is a scalar matrix. Since $Z$ is invertible, that scalar is nonzero, and $Y=u\cdot X$.

Theorem 12.1.12 Gauge uniqueness from equality of MPV families

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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 $Y=u\cdot X$.

Proof ▶

This is exactly Lemma 12.1.11; the extra MPV hypothesis specifies the setting in which the two gauges are usually produced.

Definition 12.1.13 Scalar 2-cochain

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

Definition 12.1.14 Projective representation

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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 $g,h \in G$,

\begin{align} \rho (g) \rho (h) & =\omega (g,h) \rho (gh). \label{eq:symmetry_projective_law} \end{align}

Lemma 12.1.15 Associativity forces the cocycle condition

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

\begin{align} \omega (g,h) \omega (gh,k) & =\omega (g,hk) \omega (h,k). \label{eq:symmetry_cocycle_identity} \end{align}

Proof ▶

Expand $\rho (g)\, (\rho (h)\rho (k))$ and $(\rho (g)\rho (h))\rho (k)$ using the projective multiplication law (??) and equate the results via matrix associativity. The hypothesis $D {\gt} 0$ allows cancellation of the invertible matrix factor, giving (??).

Theorem 12.1.16 Virtual representation theorem for injective MPS

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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:

a scalar 2-cochain $\omega : G \times G \to \mathbb {C}^{\times }$, and
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\} $,

\begin{align} \widetilde{A}_g^i & =X(g)\, A^i\, X(g)^{-1}. \label{eq:symmetry_virtual_gauge} \end{align}

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

\begin{align} & \begin{tenkz} [physical=up] \tnX{X(g)} & \tn[up=$i$]{A} & \tnX{X(g)^{-1}} \end{tenkz} \notag \end{align}

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 $\mathbb {C}^D$.

Proof ▶

Since $A$ is injective and on-site symmetric, each twisted tensor $\widetilde{A}_g$ is gauge equivalent to $A$ (Lemma 12.1.8). Choose $X(g) \in \mathrm{GL}_{D}(\mathbb {C})$ satisfying (??) for all $i$.

By functoriality (Lemma 12.1.3), the product $X(h)\, X(g)$ also intertwines $A$ with $\widetilde{A}_{gh}$. Since $X(gh)$ does the same, gauge uniqueness (Lemma 12.1.11) gives a nonzero scalar $\omega '(h,g)$ such that

\begin{align} X(h)\, X(g) & =\omega ’(h,g)\, X(gh). \label{eq:symmetry_gauge_factor} \end{align}

Defining $\rho (g):=X(g^{-1})$ converts (??) to the standard law (??), where $\omega (g,h):=\omega '(g^{-1},h^{-1})$.

Corollary 12.1.17 2-cocycle identity for the factor system

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If $D {\gt} 0$, the factor system $\omega $ from Theorem 12.1.16 satisfies the 2-cocycle identity for all $g,h,k \in G$:

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

Proof ▶

Theorem 12.1.16 gives $\rho $ and $\omega $. By Lemma 12.1.15, the cocycle identity (??) follows from associativity of matrix multiplication and invertibility of the representation matrices.

12.2 Cohomology Classes of Virtual Symmetry Cocycles

Two cocycles may differ by a coboundary yet represent the same projective-representation class. The following definitions and results make this precise and establish the quotient $H^2(G,\mathbb {C}^{\times })$.

Definition 12.2.1 Coboundary

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

\begin{align} \omega (g,h) & =\varphi (g)\varphi (h)\varphi (gh)^{-1}. \label{eq:symmetry_coboundary} \end{align}

Definition 12.2.2 Cohomologous cocycles

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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 $g,h \in G$,

\begin{align} \omega _1(g,h) & =\varphi (g)\varphi (h)\varphi (gh)^{-1}\omega _2(g,h). \label{eq:symmetry_cohomologous} \end{align}

Theorem 12.2.3 Gauge independence of the cocycle class

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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.

Proof ▶

Gauge uniqueness gives nonzero scalars $f(k)$ with $X_2(k)=f(k)\, X_1(k)$ for every physical twist $k$. Since the projective representation is $\rho _a(g)=X_a(g^{-1})$, setting $k=g^{-1}$ gives $\rho _2(g)=f(g^{-1})\rho _1(g)$. Equivalently, for each fixed $g$ the two local gauges

$\begin{tenkz}[physical=up] \tnX{X_1(g^{-1})} & \tn[up=$i$]{A} & \tnX{X_1(g^{-1})^{-1}} \end{tenkz}$ and $\begin{tenkz}[physical=up] \tnX{X_2(g^{-1})} & \tn[up=$i$]{A} & \tnX{X_2(g^{-1})^{-1}} \end{tenkz}$

describe the same twisted tensor, so they differ by a scalar. Set $\alpha (g):=f(g^{-1})$. Substituting $\rho _2(g)=\alpha (g)\rho _1(g)$ into the projective law (??) and cancelling the invertible matrix $X_1((gh)^{-1})=\rho _1(gh)$ yields

\begin{align} \omega _2(g,h) & =\alpha (g)\alpha (h)\alpha (gh)^{-1}\omega _1(g,h). \label{eq:symmetry_gauge_coboundary} \end{align}

Thus $\omega _2$ is the coboundary multiple of $\omega _1$ with cochain $\alpha $. This is precisely the orientation required by (??) for the stated relation $\omega _2\sim \omega _1$. Equivalently,

\begin{align} \omega _1(g,h) & =\alpha (g)^{-1}\alpha (h)^{-1}\alpha (gh)\omega _2(g,h), \notag \end{align}

so the inverse cochain also gives the symmetric conclusion $\omega _1\sim \omega _2$.

Definition 12.2.4 Multiplicative 2-cocycle condition

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

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

Theorem 12.2.5 A projective factor system is a cocycle

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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.

Proof ▶

Associativity gives the scalar identity of Lemma 12.1.15. Because the inclusion $\mathbb {C}^\times \hookrightarrow \mathbb {C}$ is injective, the same identity holds in $\mathbb {C}^\times $, which is precisely Definition 12.2.4.

Definition 12.2.6 Second cohomology quotient

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The second cohomology set is the quotient

\begin{align} H^2(G,\mathbb {C}^{\times }) & :=Z^2(G,\mathbb {C}^{\times })/{\sim }, \label{eq:symmetry_h2_quotient} \end{align}

where $\sim $ is the coboundary-equivalence relation.

Definition 12.2.7 Agreement of factor-system cohomology classes

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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.

Theorem 12.2.8 Factor-system class agreement

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

\begin{align} \rho _1 \sim _{\mathrm{proj}} \rho _2 & \iff \omega _1 \sim \omega _2. \label{eq:symmetry_projective_equivalence} \end{align}

Proof ▶

This follows from the definition of projective equivalence as cohomology-class equality for factor systems.

12.2.1 Non-triviality via the commutator phase

For an abelian symmetry group, the antisymmetric commutator phase is invariant under a change of cocycle representative. A value not equal to $1$ therefore witnesses non-triviality; no converse is asserted here.

Definition 12.2.1.1 Commutator phase

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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.

Theorem 12.2.1.2 The commutator phase is a class invariant

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

Proof ▶

Write $\omega _1(g,h)=\varphi (g)\varphi (h)\varphi (gh)^{-1}\omega _2(g,h)$ for a coboundary witness $\varphi $. Substituting into the ratio and using $gh=hg$ and the commutativity of $\mathbb {C}^{\times }$ gives

\begin{align} \beta _{\omega _1}(g,h) & = \frac{\varphi (g)\varphi (h)\varphi (gh)^{-1} \omega _2(g,h)}{\varphi (h)\varphi (g)\varphi (hg)^{-1} \omega _2(h,g)} \notag \\ & =\frac{\omega _2(g,h)}{\omega _2(h,g)} =\beta _{\omega _2}(g,h), \label{eq:symmetry_comm_phase_invariance} \end{align}

where the $\varphi $-factors cancel since $\varphi (gh)=\varphi (hg)$.

Definition 12.2.1.3 Non-trivial cohomology class

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

Theorem 12.2.1.4 Commutator-phase non-triviality test

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

Proof ▶

The trivial cocycle has commutator phase $1$ at every pair, so by Theorem 12.2.1.2 and (??), any cocycle cohomologous to it has $\beta =1$ at every commuting pair. Contrapositively, $\beta _\omega (g,h)\neq 1$ for one commuting pair forces $\omega $ to be non-cohomologous to the trivial cocycle.

12.3 Physical Endpoint String Order and Virtual-Boundary Nondecay

The source notion of string order uses physical endpoint operators and a nonidentity unitary string [ PGWS$^{+}$08 ] . We also study a restricted virtual-boundary nondecay predicate and a virtual local-covariance relation. These notions are stated separately below; equivalence with physical state symmetry requires additional hypotheses.

Definition 12.3.1 Twisted transfer map

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

\begin{align} \mathcal{E}_u(X) & := \sum _{n,n'} \langle n’| u |n\rangle A_n\, X\, A_{n'}^\dagger . \label{eq:symmetry_twisted_transfer} \end{align}

Diagrammatically one inserts the physical operator $u$ on the doubled local transfer cell:

\begin{align} & \begin{tenkz} [rows={ket,op:none,bra}] \tn{A_n} \\ \tn[role=operator]{u} \\ \tn*{A_{n'}} \end{tenkz}. \notag \end{align}

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

Definition 12.3.2 Stationary-boundary block-twist functional

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The stationary-boundary block-twist functional is

\begin{align} T_L(A,u,\Lambda ) & := \operatorname{tr}\! (\Lambda \cdot \mathcal{E}_u^L(\mathbb {1})). \label{eq:symmetry_string_parameter} \end{align}

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

\begin{align} & \begin{tenkz} [rows={ket,op:none,bra}, west={cup=$\Lambda$}, east=cup] \tn{A} & \tn{A} & \tndots & \tn{A} \\ \tn[role=operator]{u} & \tn[role=operator]{u} & \tndots & \tn[role=operator]{u} \\ \tn*{A}\tnspan[brace below]{4}{$L\text{ sites}$} & \tn*{A} & \tndots & \tn*{A} \end{tenkz}. \notag \end{align}

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.

Definition 12.3.3 Periodic finite-chain symmetry overlap

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

\begin{align} R_L(u) & :=\langle \Psi _L(A)|u^{\otimes L}|\Psi _L(A)\rangle . \label{eq:symmetry_source_periodic_RL} \end{align}

Identifying this periodic overlap with the normalized stationary-boundary functional $T_L(A,u,\Lambda )$ requires a thermodynamic-boundary and normalization hypothesis. Thus Definition 12.3.2 must not be read as the literal source definition of $R_L(u)$.

Definition 12.3.4 Physical string correlator

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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

\begin{align} S_N(x,y,u) & := \operatorname{tr}\! (\Lambda \, \mathcal{E}_x\, \mathcal{E}_u^N(\mathcal{E}_y(\mathbb {1}))). \label{eq:symmetry_physical_string} \end{align}

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

Definition 12.3.5 Fixed-endpoint physical string order

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

Definition 12.3.6 Physical endpoint string order

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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 12.3.5 holds. The comparison with arbitrary virtual boundary matrices is a separate endpoint- realizability statement.

Lemma 12.3.7 Twisted spectral-radius bound and peripheral rigidity

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 $u$,

\begin{align} \rho (\mathcal{E}_u) & \le 1. \notag \end{align}

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 $u=\sum _j e^{i\theta _j}|\widetilde j\rangle \langle \widetilde j|$:

\begin{align} V^\dagger \widetilde A_j & =e^{i(\theta -\theta _j)}\widetilde A_jV^\dagger . \label{eq:pgwsvc08_lemma1_condition} \end{align}

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

Proof ▶

Let $\mathcal{E}_u(V)=\lambda V$. Multiplication by $\Lambda V^\dagger $, the trace, and Cauchy–Schwarz give

\begin{align} |\lambda |\operatorname{tr}(V\Lambda V^\dagger ) & \le \left(\sum _j\operatorname{tr}(V\widetilde A_j^\dagger \Lambda \widetilde A_jV^\dagger )\right)^{1/2} \left(\sum _j\operatorname{tr}(\widetilde A_j^\dagger V\Lambda V^\dagger \widetilde A_j)\right)^{1/2} \\ & =\operatorname{tr}(V\Lambda V^\dagger ). \notag \end{align}

Faithfulness of $\Lambda $ yields $|\lambda |\le 1$. Equality in Cauchy–Schwarz is equivalent to (??); conversely that relation makes $e^{i\theta }$ an eigenvalue. It also gives $\mathcal{E}_A(V^\dagger V)=V^\dagger V$. The fixed space of the pure canonical transfer map is one-dimensional, so $V^\dagger V=c\mathbb {1}$ for some $c{\gt}0$. Replacing $V$ by $c^{-1/2}V$ makes it unitary without changing the intertwining relation.

If $V,V'$ are peripheral eigenmatrices with eigenvalues $e^{i\theta },e^{i\theta '}$, respectively, the equality relations give the exact identity

\begin{align} \mathcal{E}_A(V^\dagger V’) & =e^{i(\theta '-\theta )}V^\dagger V’. \notag \end{align}

Purity leaves only the peripheral eigenvalue $1$, hence $\theta '=\theta $ modulo $2\pi $. The one-dimensional fixed space then gives $V^\dagger V'\in \mathbb {C}\mathbb {1}$, proving uniqueness of the peripheral eigenvalue and its eigendirection.

Theorem 12.3.8 Physical endpoint criterion for string order

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

\begin{align} \mathcal{E}_{\widetilde u}(V) & =V, & \operatorname{tr}(V\Lambda A_nA_m^\dagger ) & \ne 0. \label{eq:pgwsvc08_theorem1} \end{align}

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.

Proof ▶

Lemma 12.3.7 shows that a nonzero limit requires a unique peripheral eigenvalue $e^{i\theta }$. If $V$ and $Y$ are the corresponding right and left eigenmatrices, then $Y=\Lambda V^\dagger $, and the leading term retains its length-dependent phase:

\begin{align} S_N(x,y,u) & =e^{iN\theta } \operatorname{tr}\! \bigl(Y\mathcal{E}_y(\mathbb {1})\bigr) \operatorname{tr}\! \bigl(\Lambda \mathcal{E}_x(V)\bigr)+o(1). \notag \end{align}

In the eigenbasis $u=\sum _j e^{i\theta _j}|\widetilde j\rangle \langle \widetilde j|$, shift the string unitary by the peripheral phase,

\begin{align} \widetilde u & :=e^{-i\theta }u =\sum _j e^{i(\theta _j-\theta )} |\widetilde j\rangle \langle \widetilde j|. \notag \end{align}

Then $\mathcal{E}_{\widetilde u}(V)=V$. Choosing $y=x^\dagger \widetilde u$ reduces nonvanishing to the second condition in (??); if it holds for $n,m$, take $x=|n\rangle \langle m|$. The reverse implication follows from the same phase-retaining factorization and the endpoint-realizability hypothesis.

Theorem 12.3.9 Physical state symmetry and twisted spectral radius

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

\begin{align} \rho (\mathcal{E}_u) & =1. \notag \end{align}

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.

Proof ▶

If $\rho (\mathcal{E}_u)=1$, Lemma 12.3.7 gives a unitary $V$ satisfying the local intertwining relation. Substitution of that relation into the adjoint transfer map, together with $\mathcal{E}_A^\dagger (\Lambda )=\Lambda $, gives $\mathcal{E}_A^\dagger (V\Lambda V^\dagger )=V\Lambda V^\dagger $. This positive fixed point has trace one, so uniqueness gives $V\Lambda V^\dagger =\Lambda $, equivalently $V^\dagger \Lambda V=\Lambda $, which yields the physical symmetry. Conversely, physical invariance gives, for length-independent boundary tensors $L,R$,

\begin{align} D^{-2} & \le \operatorname{tr}(\varrho _N^2) =\operatorname{tr}\! \left[L(\mathcal{E}_u\otimes \mathcal{E}_{u^\dagger })^N(R)\right]. \notag \end{align}

Exponential decay would contradict this bound, so $\rho (\mathcal{E}_u)=1$.

Definition 12.3.10 Virtual local covariance with boundary invariance

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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 $i$,

\begin{align} \sum _j u_{ij}\, A^j & = \mu \, V\, A^i\, V^\dagger . \label{eq:symmetry_local_covariance} \end{align}

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

Definition 12.3.11 Virtual-boundary nondecay

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

\begin{align} R_L(u;X,Y) & :=\operatorname{tr}\! \bigl(\Lambda X\mathcal{E}_u^L(Y)\bigr). \notag \end{align}

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 12.3.6 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.

Definition 12.3.12 Local physical intertwining

✓

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:

$\begin{tenkz}[rows={op:none,ket}] \tn[role=operator,up=$i$]{u} \\ \tn{A} \end{tenkz}$ = $\begin{tenkz}[physical=up] \tnX{V} & \tn[up=$i$]{A} & \tnX{V^\dagger} \end{tenkz}$ .

Definition 12.3.13 Transfer-map covariance

✓

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

\begin{align} \mathcal{E}(V X V^\dagger ) & = V\, \mathcal{E}(X)\, V^\dagger . \label{eq:symmetry_transfer_covariance} \end{align}

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

$\begin{tenkz}[sandwich] \tnX{V} & \tn{A} \\ \tnX{\overline V} & \tn*{A} \end{tenkz}$ = $\begin{tenkz}[sandwich] \tn{A} & \tnX{V} \\ \tn*{A} & \tnX{\overline V} \end{tenkz}$ .

Definition 12.3.14 Commutation in the doubled transfer picture

✓

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

\begin{align} V\, \mathcal{E}_1(X)\, V^\dagger & = \mathcal{E}_1(V X V^\dagger ). \label{eq:symmetry_doubled_commutation} \end{align}

Theorem 12.3.15 Transfer-map covariance and doubled commutation are equivalent

✓

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

Proof ▶

Both sides express the same local picture with opposite orientation:

$\begin{tenkz}[sandwich] \tnX{V} & \tn{A} \\ \tnX{\overline V} & \tn*{A} \end{tenkz}$ = $\begin{tenkz}[sandwich] \tn{A} & \tnX{V} \\ \tn*{A} & \tnX{\overline V} \end{tenkz}$ .

The two formulas are the same identity read from opposite sides.

Theorem 12.3.16 Local physical intertwining implies transfer-map covariance

✓

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

Proof ▶

Starting from

$\begin{tenkz}[rows={op:none,ket}] \tn[role=operator,up=$i$]{u} \\ \tn{A} \end{tenkz}$ = $\begin{tenkz}[physical=up] \tnX{V} & \tn[up=$i$]{A} & \tnX{V^\dagger} \end{tenkz}$ ,

one inserts $V^\dagger V = \mathbb {1}$ so that each local transfer cell is written in terms of conjugated Kraus operators, then replaces those operators using the intertwining relation. After that, the physical $u$-boxes disappear by the unitary Kraus-mixing identity.

Theorem 12.3.17 Transfer-map covariance implies local physical intertwining

✓

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.

Proof ▶

Set $B^i := V\, A^i\, V^\dagger $. The transfer-map covariance relation

$\begin{tenkz}[sandwich] \tnX{V} & \tn{A} \\ \tnX{\overline V} & \tn*{A} \end{tenkz}$ = $\begin{tenkz}[sandwich] \tn{A} & \tnX{V} \\ \tn*{A} & \tnX{\overline V} \end{tenkz}$

implies that the two $d$-element Kraus families $B$ and $A$ define the same transfer map. Rectangular Kraus freedom (Theorem 4.5.10) converts $\mathcal{E}_B = \mathcal{E}_A$ into a unitary Kraus mixing matrix $u$ with $B^i = \sum _j u_{ij} A^j$.

Theorem 12.3.18 Twisted-transfer eigenvalues have modulus at most one

✓

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 $|\lambda | \le 1$.

Proof ▶

By Lemma H.4.2, the twisted transfer map is the mixed transfer operator $F_{A,B_u}$, as in (??). Because $B_u$ is obtained from $A$ by unitary mixing of the Kraus operators, the transfer maps of $A$ and $B_u$ coincide, so the two adjoint channels share a common positive definite fixed point. Gauging both families by that fixed point puts them in a common trace-preserving gauge. Theorem 7.4.1 then applies to the mixed transfer operator in that gauge, and similarity preserves the eigenvalue $\lambda $.

Theorem 12.3.19 Modulus-one twisted eigenvalue implies gauge-phase equivalence

✓

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

\begin{align} \mathcal{E}_u(X) & = \lambda X \label{eq:symmetry_twisted_eigen_mod_one} \end{align}

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

Proof ▶

By Lemma H.4.2, the twisted transfer map is the mixed transfer operator $F_{A,B_u}$. Because $B_u$ is obtained from $A$ by unitary mixing of the Kraus operators, the transfer maps of $A$ and $B_u$ coincide, so the two adjoint channels share a common positive definite fixed point. Gauging both families by that fixed point puts them in a common trace-preserving gauge. In that gauge, (??) becomes a modulus-one peripheral eigenvalue for an irreducible mixed transfer operator, so Theorem 7.7.1 yields gauge-phase equivalence. Undoing the common trace-preserving gauge gives the claim for $A$ and $B_u$.

Theorem 12.3.20 Virtual-boundary nondecay gives companion gauge-phase equivalence

✓

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 $B_u$.

Proof ▶

If $\rho (\mathcal{E}_u) {\lt} 1$, Lemma H.6.1 forces every virtual-boundary sequence to decay to zero, contradicting Definition 12.3.11. Hence $\rho (\mathcal{E}_u) \ge 1$. Theorem 12.3.18 shows that every eigenvalue has modulus at most $1$, so some peripheral eigenvalue has modulus exactly $1$. Applying Theorem 12.3.19 then gives the gauge-phase equivalence.

Theorem 12.3.21 Gauge-phase equivalence with the companion tensor yields a virtual unitary

✓

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 $i$,

\begin{align} \sum _j u_{ij} A^j & = \mu \, V A^i V^\dagger . \label{eq:symmetry_phased_covariance} \end{align}

Proof ▶

Start from an invertible intertwiner $X$ and a nonzero gauge scalar $\zeta $ between $A$ and $B_u$. The positive matrix $Q:=XX^\dagger $ satisfies $\mathcal{E}_A(Q)=|\zeta |^2Q$. Since $\mathcal{E}_A(\mathbb {1})=\mathbb {1}$, irreducibility forces the two positive eigenvalues to agree, so $|\zeta |=1$ and $Q$ is a positive fixed point of $\mathcal{E}_A$. Uniqueness of positive fixed points for an injective normalized tensor then makes $Q$ a positive scalar multiple of the identity. Rescaling $X$ by the square root of that scalar produces a unitary $V$, and the original intertwining relation becomes (??), with $|\mu |=1$.

Theorem 12.3.22 Virtual unitary gives a twisted-transfer eigenmatrix

✓

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:

\begin{align} \mathcal{E}_u(V) & = \mu V. \label{eq:symmetry_virtual_eigenmatrix} \end{align}

Proof ▶

Expand $\mathcal{E}_u(V)$ using (??), insert the intertwining relation into each local term, and use $V^\dagger V = \mathbb {1}$ together with $\mathcal{E}_A(\mathbb {1})=\mathbb {1}$.

Theorem 12.3.23 Boundary-state invariance of the virtual unitary

✓

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 $i$,

\begin{align} \sum _j u_{ij} A^j & = \mu \, V A^i V^\dagger , \label{eq:symmetry_boundary_covariance} \end{align}

then

\begin{align} V^\dagger \Lambda V & = \Lambda . \label{eq:symmetry_boundary_invariance} \end{align}

Proof ▶

Relation (??) first shows that $V\Lambda V^\dagger $ is another positive fixed point of the adjoint transfer map. It has the same trace as $\Lambda $, so uniqueness of the normalized positive fixed point gives $V\Lambda V^\dagger =\Lambda $. Multiplying by $V^\dagger $ on the left and by $V$ on the right yields (??).

Lemma 12.3.24 Virtual local covariance implies virtual-boundary nondecay

✓

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 $u$.

Proof ▶

Choose the boundary matrices $X = V^\dagger $ and $Y = V$, where $V$ is the unitary from the virtual local-covariance datum. By Theorem 12.3.22, one has $\mathcal{E}_u^L(V)=\mu ^L V$. Therefore

\begin{align} R_L(u;V^\dagger ,V) & = \mu ^L \operatorname{tr}(\Lambda ) = \mu ^L, \notag \end{align}

whose norm is constantly $1$.

Theorem 12.3.25 Virtual local covariance iff a unitary peripheral eigenmatrix exists

✓

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 12.3.18, this witness form is equivalent to the peripheral-spectrum condition $\rho (\mathcal{E}_u)=1$.

Proof ▶

The forward direction is Theorem 12.3.22. For the reverse direction, start from a unitary peripheral eigenmatrix. Theorem 12.3.19 gives gauge-phase equivalence with the companion tensor; then Theorem 12.3.21 yields a virtual unitary and phase satisfying the local covariance relation, and Theorem 12.3.23 gives the boundary-state invariance needed for virtual local covariance.

Theorem 12.3.26 Virtual-boundary nondecay $\Leftrightarrow $ virtual local covariance

✓

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.

Proof ▶

If virtual-boundary nondecay holds, Theorem 12.3.20 gives a gauge-phase equivalence with the companion tensor. Theorem 12.3.21 extracts a virtual unitary and phase from that equivalence, and Theorem 12.3.23 shows that the same unitary preserves the boundary state, so one obtains virtual local covariance. Conversely, Lemma 12.3.24 turns virtual local covariance into a nondecaying virtual-boundary sequence.

Theorem 12.3.27 Virtual unitary from virtual-boundary nondecay

✓

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:

$\begin{tenkz}[rows={op:none,ket}] \tn[role=operator,up=$i$]{u} \\ \tn{A} \end{tenkz}$ $\; =\; \mu \, $ $\begin{tenkz}[physical=up] \tnX{V} & \tn[up=$i$]{A} & \tnX{V^\dagger} \end{tenkz}$ .

Proof ▶

Virtual-boundary nondecay first gives gauge-phase equivalence with the companion tensor by Theorem 12.3.20. Theorem 12.3.21 then normalizes the resulting intertwiner to a unitary and produces the phase $\mu $.

12.4 SPT Phase Labels and Virtual-Boundary Nondecay

The predicate in Definition 12.4.1 compares virtual cocycle labels. The source classification below concerns symmetric paths that remain within the MPS and corresponding parent-Hamiltonian framework, with the endpoint representations embedded into a common physical space as in [ SPGC11 ] .

Definition 12.4.1 Equality of virtual cocycle labels

✓

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

\begin{align} \widetilde A_g^i & =\rho _A(g^{-1})A^i\rho _A(g^{-1})^{-1}, \\ \widetilde B_g^i & =\rho _B(g^{-1})B^i\rho _B(g^{-1})^{-1}, \notag \end{align}

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.

Remark 12.4.2 Scalar and character conventions

These factor systems take values in $\mathbb {C}^\times $. For unitary bond representations one may choose unit-modulus representatives and obtain the physical classification label in $H^2(G,\mathrm U(1))$; this normalization is not an equality $\mathbb {C}^\times =\mathrm U(1)$. Moreover, Definition 12.4.1 requires exact tensor equations. A physical state symmetry may instead act on an MPS vector by a one-dimensional character $\chi (g)$, producing a scalar factor in the local tensor equation. Exact MPV-family equality and such a character are distinct hypotheses.

Theorem 12.4.3 Classification along symmetric MPS parent-Hamiltonian paths

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

\begin{align} \mathcal{H} & =\mathcal{H}_0\oplus \mathcal{H}_1\oplus \mathcal{H}_{\mathrm{path}}, & U_g & =U_g^0\oplus U_g^1\oplus U_g^{\mathrm{path}}. \notag \end{align}

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.

Proof ▶

First deform each blocked injective MPS to its isometric form; the deformation preserves the symmetry relation

\begin{align} U_g\mathcal{P} & =\mathcal{P}(V_g\otimes \overline{V_g}). \notag \end{align}

Along any continuous symmetric MPS path quantified in the theorem, the tensor and its induced virtual action $V_g$ vary continuously. The cohomology class is discrete, so it cannot change along that path. This proves necessity within the MPS/parent-Hamiltonian framework.

Conversely, let the endpoint isometric forms have virtual actions $V_g^0$ and $V_g^1$ in the same cohomology class. Embed their bond spaces into the direct sum and interpolate between the maximally entangled bonds by

\begin{align} |\omega (\gamma )\rangle & =(1-\gamma )\sum _{i=1}^{D_0}|i,i\rangle +\gamma \sum _{i=D_0+1}^{D_0+D_1}|i,i\rangle . \notag \end{align}

The commuting parent Hamiltonians along this interpolation remain gapped. Their symmetry is

\begin{align} (V_g^0\oplus V_g^1)\otimes (\overline{V_g^0}\oplus \overline{V_g^1}) & =\widehat U_g^0\oplus \widehat U_g^1 \oplus \widehat U_g^{\mathrm{path}}, \\ \widehat U_g^{\mathrm{path}} & =(V_g^0\otimes \overline{V_g^1}) \oplus (V_g^1\otimes \overline{V_g^0}). \notag \end{align}

Equality of the cohomology classes is exactly what makes the cross-term action $\widehat U^{\mathrm{path}}$ a linear representation. Conjugating back by the endpoint isometries gives the common direct-sum physical representation and the required symmetric gapped path.

Theorem 12.4.4 Virtual-boundary nondecay for injective MPV-symmetric tensors

✓

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 $g$.

Proof ▶

The virtual representation theorem produces $\rho (g^{-1}) \in \mathrm{GL}_{D}(\mathbb {C})$ satisfying $\sum _j U(g)_{ij}\, A^j = \rho (g^{-1})\, A^i\, \rho (g^{-1})^{-1}$. A direct computation using $\mathcal{E}_A(\mathbb {1})=\mathbb {1}$ shows that $\mathcal{E}_{U(g)}(\rho (g^{-1}))=\rho (g^{-1})$, so $\rho (g^{-1})$ is a nonzero eigenvector with eigenvalue $1$. Theorem 12.3.19 gives gauge-phase equivalence with the companion tensor, and Theorem 12.3.21 extracts a virtual unitary and phase. By Theorem 12.3.23, this unitary preserves the boundary state, so Lemma 12.3.24 gives virtual-boundary nondecay.

Remark 12.4.5 Virtual-boundary nondecay does not determine the SPT label

Theorem 12.4.4 applies to every injective symmetric tensor with canonical normalisations, irrespective of its cocycle class, so tensors in different SPT phases all satisfy virtual-boundary nondecay for every group element. This nondecay is equivalent to virtual local covariance (Theorem 12.3.26); the invariant that separates SPT phases is the cohomology class of the virtual representation cocycle (Theorem 12.4.3).