A ${{\mathbb {Z}}}_2$-Index of Symmetry Protected Topological Phases with Time Reversal Symmetry for Quantum Spin Chains

Abstract

We introduce a ${{\mathbb {Z}}}_2$-index for time reversal invariant Hamiltonians with unique gapped ground state on quantum spin chains. We show this is an invariant of a $C^1$-classification of gapped Hamiltonians. Analogous results hold for more general on-site finite group symmetry, with the 2-cohomology class as the invariant.

Appendices

Proof of Lemma 5.1 and Lemma 5.2

In this section we prove Lemma 5.1 and Lemma 5.2. The proof is based on arguments and tools in [BMNS]. For M in Condition B, we may and will assume that $M>2$ . Let us first recall the Lieb–Robinson bound. Fix some $a>0$ (throughout this appendix), and define a positive function $F_a(r)$ on ${{\mathbb {R}}}_{\ge 0}$ by $F_a(r):=(1+r)^{-2}e^{-ar}$. For a path of interactions satisfying Definition 3.4, there exist positive constants $C_{1,a}$, $v_a$ satisfying the following.: For any $X,Y\in {{\mathfrak {S}}}_{{{\mathbb {Z}}}}$, $A\in {{\mathcal {A}}}_X$, $B\in {{\mathcal {A}}}_Y$, $k\in {\mathbb {N}}$, $s\in [0,1]$ and $t\in {{\mathbb {R}}}$, we have

$$\begin{aligned}&\left\| \left[ \tau _{t}^{\Phi (s)}(A),B \right] \right\| , \left\| \left[ \tau _{t}^{{\tilde{\Phi }}(s)}(A),B \right] \right\| , \left\| \left[ \tau _{t}^{\Phi (s),\Lambda _{n_k}}(A),B \right] \right\| , \nonumber \\&\left\| \left[ \tau _{t}^{{\tilde{\Phi }}(s),\Lambda _{n_k}}(A),B \right] \right\| , \left\| \left[ \tau _{t}^{\Phi (s)+\Psi _{k}(s),\Lambda _{n_k}}(A),B \right] \right\| \nonumber \\&\quad \le C_{1,a}e^{v_a|t|} \sum _{x\in X,y\in Y}F_a(|x-y|)\left\| A\right\| \left\| B\right\| . \end{aligned}$$

(57)

(The inequality means that each of the left hand side can be bounded by the same value written on the right hand side. We use this way of writing below as well.) As in the proof of Theorem 2.2 [NOS], perturbation of dynamics can be estimated by the use of the Lieb–Robinson bound. In particular, by the Lieb–Robinson bound (57) and 2. of Defintion 3.4, for the fixed $a>0$ above, there exists a constant $C_{2,a}$ such that

$$\begin{aligned}&\left\| \tau _t^{{\tilde{\Phi }}(s),\Lambda _{n}}\left( A\right) -\tau _t^{\Phi (s),\Lambda _{n}}\left( A\right) \right\| \nonumber \\&\quad =\left\| \int _0^t du \frac{d}{du}\left( \tau _{t-u}^{{\tilde{\Phi }}(s),\Lambda _{n}}\circ \tau _u^{\Phi (s),\Lambda _{n}}\left( A\right) \right) \right\| \nonumber \\&\quad =\left\| \int _0^t du \;\tau _{t-u}^{{\tilde{\Phi }}(s),\Lambda _{n}}\left( \sum _{\begin{array}{c} X\cap [0,\infty )\ne \emptyset ,\; X\cap (-\infty ,-1]\ne \emptyset \end{array}} i\left[ \Phi (X;s),\tau _u^{\Phi (s),\Lambda _{n}}\left( A\right) \right] \right) \right\| \nonumber \\&\quad \le C_{2,a}\sum _{y\in Y}e^{v_a|t|-a|y|}\left\| A\right\| , \end{aligned}$$

(58)

for all $t\in {{\mathbb {R}}}$, $s\in [0,1]$, $n\in {\mathbb {N}}$, $Y\in {{\mathfrak {S}}}_{{{\mathbb {Z}}}}$, and $A\in {{\mathcal {A}}}_Y$. Here, $v_a$ is the same constant as in (57). Similarly, we have

$$\begin{aligned}&\left\| \tau _t^{\Phi (s),\Lambda _{n_k}}\left( A\right) -\tau _t^{\Phi (s)+\Psi _k(s),\Lambda _{n_k}}\left( A\right) \right\| \le C_{3,a}\sum _{y\in Y}e^{v_a|t|-a\cdot d(y,\left( \Lambda _{n_k-R}\right) ^c)}\left\| A\right\| , \end{aligned}$$

(59)

for all $t\in {{\mathbb {R}}}$, $s\in [0,1]$, $k\in {\mathbb {N}}$, $Y\in {{\mathfrak {S}}}_{{{\mathbb {Z}}}}$, and $A\in {{\mathcal {A}}}_Y$.

Taking $n\rightarrow \infty$ limit in (58), we obtain

$$\begin{aligned} \left\| \tau _t^{{\tilde{\Phi }}(s)}\left( A\right) -\tau _t^{\Phi (s)}\left( A\right) \right\| \le C_{2,a}\sum _{y\in Y}e^{v_a|t|-a|y|}\left\| A\right\| , \end{aligned}$$

(60)

for all $t\in {{\mathbb {R}}}$, $s\in [0,1]$, $Y\in {{\mathfrak {S}}}_{{{\mathbb {Z}}}}$, and $A\in {{\mathcal {A}}}_Y$. This estimate tells us that if A is far away from the origin of ${{\mathbb {Z}}}$ compared to |t|, the difference between the dynamics given by $\Phi (s)$ and ${\tilde{\Phi }}(s)$ is small.

By the same argument as in (58), for the fixed $a>0$, there exists a positive constant $C_{3,a}$ such

$$\begin{aligned}&\left\| \tau _t^{{\tilde{\Phi }}(s),\Lambda _{m}}\left( A\right) -\tau _t^{{\tilde{\Phi }}(s),\Lambda _{n}}\left( A\right) \right\| ,\; \left\| \tau _t^{\Phi (s),\Lambda _{m}}\left( A\right) -\tau _t^{\Phi (s),\Lambda _{n}}\left( A\right) \right\| ,\nonumber \\&\left\| \tau _t^{{\tilde{\Phi }}(s),\Lambda _{m}}\left( A\right) -\tau _t^{{\tilde{\Phi }}(s)}\left( A\right) \right\| ,\; \left\| \tau _t^{\Phi (s),\Lambda _{m}}\left( A\right) -\tau _t^{\Phi (s)}\left( A\right) \right\| \nonumber \\&\quad \le C_{3,a}e^{v_a|t|} \sum _{y\in Y}\sum _{x\in \Lambda _{m}^c} F_a\left( |x-y| \right) \left\| A\right\| \end{aligned}$$

(61)

for all $n,m\in {\mathbb {N}}$, $n>m$, $t\in {{\mathbb {R}}}$, $s\in [0,1]$, $Y\in {{\mathfrak {S}}}_{{{\mathbb {Z}}}}$, and $A\in {{\mathcal {A}}}_Y$. Here, $v_a$ is the same constant as in (57). For each $k\in {\mathbb {N}}$, we denote by $m_k$ the the smallest integer less than or equal to $n_k/2$.

Proof of Lemma 5.1

We first show that

$$\begin{aligned} \lim _{k\rightarrow \infty } \left\| \left( {\hat{\alpha }}_{s}^{(k,o)}\right) ^{-1}(A)-\left( \alpha _s^{(k)}\right) ^{-1}(A) \right\| =0, \end{aligned}$$

(62)

for any $l\in {\mathbb {N}}$ and $A\in {{\mathcal {A}}}_{\Lambda _l}$. Fix any $l\in {\mathbb {N}}$ and $A\in {{\mathcal {A}}}_{\Lambda _l}$. We may and we will assume that $n_k\ge 4(M+R+l)$ for each $k\in {\mathbb {N}}$. For each $k\in {\mathbb {N}}$, we have

$$\begin{aligned} \frac{d}{ds}\alpha _s^{(k)}\circ \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) =\alpha _s^{(k)} \left( i\left[ -D_k(s)+{\hat{D}}_{k,o}(s),\; \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right] \right) . \end{aligned}$$

(63)

We claim

$$\begin{aligned} \varepsilon _k(A):=\sup _{s\in [0,1]}\left\| \left[ -D_k(s)+{\hat{D}}_{k,o}(s),\; \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right] \right\| \rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$

(64)

To show this, we split $\left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right)$ into two parts. For each k, we denote by $L_k$, the smallest integer less than or equal to $\frac{n_k}{4}$. Recall also that $m_k$ is the smallest integer less than or equal to $\frac{n_k}{2}$. From [BMNS] proof of Theorem 4.5 and Lemma 3.2, $\left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right)$ can be decomposed into an element $\Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right)$ in ${{\mathcal {A}}}_{L_k}$ with $\left\| \Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right) \right\| \le \left\| A\right\|$, and the rest, which is bounded from above as

$$\begin{aligned} \left\| \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) - \Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right) \right\| \le C_1\;(2l+1)\;{\tilde{u}}\left( d\left( \Lambda _l,\Lambda _{n_k}{\setminus } \Lambda _{L_k} \right) \right) \left\| A\right\| . \end{aligned}$$

(65)

The function ${\tilde{u}}(r)$, $r>0$ on the right hand side satisfies ${\tilde{u}}(r)\rightarrow 0$, as $r\rightarrow \infty$.

The difference $-D_k(s)+{\hat{D}}_{k,o}(s)$ is localized at the boundary of $\Lambda _{n_k}$. Therefore, by Lieb–Robinson bound, it almost commutes with $\Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right)$ for k large enough. For simplicity, let us introduce a notation

$$\begin{aligned} B(X,s,t,k):= \tau _{t}^{\Phi (s),\Lambda _{n_k}} \left( \Phi '(X;s)\right) -\tau _{t}^{\Phi (s)+\Psi _{k}(s),\Lambda _{n_k}} \left( \Phi '(X;s)\right) , \end{aligned}$$

(66)

for $X\in {{\mathfrak {S}}}_{{{\mathbb {Z}}}}$, $t\in {{\mathbb {R}}}$, $s\in [0,1]$, and $k\in {{\mathbb {N}}}$. We have

$$\begin{aligned} -D_k(s)+{\hat{D}}_{k,o}(s)&=\sum _{X\subset \Lambda _{n_k}} \int _{-\infty }^\infty dt \; W_\gamma (t) B(X,s,t,k) \nonumber \\&\quad -\sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X \subset \Lambda _{n_k}{\setminus } \Lambda _{n_k-R} \end{array}} \int _{-\infty }^\infty dt \; W_\gamma (t) \tau _{t}^{\Phi (s)+\Psi _{k}(s),\Lambda _{n_k}} \left( \Psi _k'(X;s)\right) . \end{aligned}$$

(67)

Set

$$\begin{aligned} T_X^k:=\frac{a}{2v_a} \cdot d\left( X,\left( \Lambda _{n_k-R}\right) ^c\right) ,\quad S_X^k:=\frac{a}{2v_a} \cdot d\left( X,\Lambda _{L_k}\right) \end{aligned}$$

(68)

for each $k\in {\mathbb {N}}$ and $X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}$. We split the summation of $X\subset \Lambda _{n_k}$ in the first term of (67) into $X\subset \Lambda _{m_k}$ and $X\cap \left( \Lambda _{m_k}\right) ^c\ne \emptyset$. For $X\subset \Lambda _{m_k}$, we split the integration into $|t|\le T_X^{k}$ part and $|t|\ge T_X^{k}$ part. For $X\cap \left( \Lambda _{m_k}\right) ^c\ne \emptyset$, we split the integration into $|t|\le S_X^{k}$ part and $|t|\ge S_X^{k}$ part.

First we consider $X\subset \Lambda _{m_k}$ and $|t|\le T_X^{k}$ part. From (59), and Definition 3.42., we have

$$\begin{aligned}&\left\| \sum _{X\subset \Lambda _{m_k}} \int _{|t|\le T_X^k} dt W_\gamma (t) B(X,s,t,k) \right\| \nonumber \\&\quad \le \sum _{\begin{array}{c} X\subset \Lambda _{m_k}\\ \mathop {\mathrm {diam}}\nolimits X<M \end{array}}\left\| W_\gamma \right\| _1 C_1C_{3,a}\sum _{y\in X}e^{v_aT_X^k-a\cdot d\left( y,\left( \Lambda _{n_k-R}\right) ^c\right) }\nonumber \\&\quad \le \sum _{\begin{array}{c} X\subset \Lambda _{m_k}\\ \mathop {\mathrm {diam}}\nolimits X<M \end{array}}\left\| W_\gamma \right\| _1 C_1C_{3,a}Me^{-\frac{a}{2}\cdot d\left( X,\left( \Lambda _{n_k-R}\right) ^c\right) } \nonumber \\&\quad =C_1C_{3,a}M\left\| W_\gamma \right\| _1\sum _{j=n_k-m_k-R}^\infty \sum _{\begin{array}{c} X\subset \Lambda _{m_k}\\ \mathop {\mathrm {diam}}\nolimits X<M\\ d\left( X,\left( \Lambda _{n_k-R}\right) ^c\right) =j \end{array}}e^{-\frac{a}{2} j}\nonumber \\&\quad \le C_1C_{3,a}M2^{M}\left\| W_\gamma \right\| _1\sum _{j=n_k-m_k-R}^\infty e^{-\frac{a}{2} j}. \end{aligned}$$

(69)

Note that for $X\subset \Lambda _{m_k}$, the distance between X and $\left( \Lambda _{n_k-R}\right) ^c$ is at least $n_k-R-m_k$. This is used in the equality in the second line. Recall that $n_k-m_k-R\ge 1$ as we assumed $n_k\ge 4(M+R+l)$ in the beginning of the proof. In the last inequality, we used the fact that for any $j\ge 1$, the number of $X\subset \Lambda _{m_k}$ with $\mathop {\mathrm {diam}}\nolimits (X)<M$ such that $d\left( X,\left( \Lambda _{n_k-R}\right) ^c\right) =j$ is at most $2^{M}$. Note that the last line of (69) is independent of $s\in [0,1]$ and goes to 0 as $k\rightarrow \infty$.

Next we estimate the first term of (67) corresponding to $X\cap \left( \Lambda _{m_k}\right) ^c\ne \emptyset$ and $|t|\le S_X^{k}$ part. The corresponding part of $-D_k(s)+{\hat{D}}_{k,o}(s)$ is not necessarily small, but it is localized at the edge of $\Lambda _{n_k}$. Therefore, the commutator with $\Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right)$ is small. From the Lieb–Robinson bound (57), by the same kind of argument as in (69)

$$\begin{aligned}&\left\| \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \end{array}} \int _{|t|\le S_X^k} dt W_\gamma (t) \left[ B(X,s,t,k),\;\Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right) \right] \right\| \nonumber \\&\quad \le \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \end{array}} \int _{|t|\le S_X^k} dt \left| W_\gamma (t)\right| \nonumber \\&\qquad \left( \begin{gathered} \left\| \left[ \tau _{t}^{\Phi (s),\Lambda _{n_k}} \left( \Phi '(X;s)\right) ,\;\Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right) \right] \right\| \\ +\left\| \left[ \tau _{t}^{\Phi (s)+\Psi _{k}(s),\Lambda _{n_k}} \left( \Phi '(X;s)\right) ,\;\Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right) \right] \right\| \end{gathered} \right) \nonumber \\&\quad \le 2C_{1,a}C_1\left\| W_\gamma \right\| _1\left\| A\right\| \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \\ \mathop {\mathrm {diam}}\nolimits X<M \end{array}} e^{v_aS_X^k} \sum _{x\in X,y\in \Lambda _{L_k}}F_a(|x-y|)\nonumber \\&\quad \le 2C_{1,a}C_1\left\| W_\gamma \right\| _1\left\| A\right\| M\sum _{y\in {{\mathbb {Z}}}}F(|y|) \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \\ \mathop {\mathrm {diam}}\nolimits X<M \end{array}} e^{v_aS_X^k-a\cdot d\left( X,\Lambda _{L_k}\right) }\nonumber \\&\quad = 2C_{1,a}C_1\left\| W_\gamma \right\| _1\left\| A\right\| M\sum _{y\in {{\mathbb {Z}}}}F(|y|) \sum _{j=m_k-M-L_k}^\infty \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \\ \mathop {\mathrm {diam}}\nolimits X<M\\ d\left( X,\Lambda _{L_k} \right) =j \end{array}} e^{v_aS_X^k-a\cdot d\left( X,\Lambda _{L_k}\right) }\nonumber \\&\quad \le 2C_{1,a}C_1\left\| W_\gamma \right\| _1\left\| A\right\| M\sum _{y\in {{\mathbb {Z}}}}F(|y|)2^{M} \sum _{j=m_k-M-L_k}^\infty e^{-\frac{aj}{2}}. \end{aligned}$$

(70)

As we assumed that k is large enough so that $n_k\ge 4(M+R+l)$, we have $m_k-M-L_k\ge 1$. Therefore, in the last inequality, the number of $X\cap \left( \Lambda _{m_k}\right) ^c\ne \emptyset$ with $\mathop {\mathrm {diam}}\nolimits X<M$ and $d(X,\Lambda _{L_k})=j\ge 1$ is bounded by $2^{M}$. The last line is independent of $s\in [0,1]$ and goes to 0 as $k\rightarrow \infty$.

For $X\cap \left( \Lambda _{m_k}\right) ^c\ne \emptyset$, and $|t|\ge S_X^{k}$ part, we have

$$\begin{aligned} \left\| \sum _{\begin{array}{c} X\cap \left( \Lambda _{m_k}\right) ^c\ne \emptyset \\ X\subset \Lambda _{n_k} \end{array}} \int _{|t|\ge S_X^k} dt W_\gamma (t) B(X,s,t,k) \right\|&\le 4C_1\sum _{\begin{array}{c} X\cap \left( \Lambda _{m_k}\right) ^c\ne \emptyset \\ X\subset \Lambda _{n_k}\\ \mathop {\mathrm {diam}}\nolimits X<M \end{array}} I_\gamma (S_{X}^k) \nonumber \\&\le 2^{M+2}C_1\sum _{j=m_k-M-L_k}^\infty I_\gamma \left( \frac{aj}{2v_a}\right) . \end{aligned}$$

(71)

In the first inequality we used $B(X,s,t,k)\le 2C_1$ and (20) and the oddness of $W_\gamma (t)$. As we assumed that k is large enough so that $n_k\ge 4(M+R+l)$, we have $m_k-M-L_k\ge 1$. Therefore, in the second inequality, the number of $X\cap \left( \Lambda _{m_k}\right) ^c\ne \emptyset$ with $\mathop {\mathrm {diam}}\nolimits X<M$ and $d(X,\Lambda _{L_k})=j\ge 1$ is bounded by $2^{M}$. The right hand side is independent of $s\in [0,1]$ and goes to 0 as $k\rightarrow \infty$.

Similary, we may estimate $X\subset \Lambda _{m_k}$ and $|t|\ge T_X^{k}$ part.

$$\begin{aligned} \left\| \sum _{X\subset \Lambda _{m_k}} \int _{|t|\ge T_X^k} dt \; W_\gamma (t) B(X,s,t,k) \right\|&\le 4C_1\sum _{\begin{array}{c} X\subset \Lambda _{m_k}\\ \mathop {\mathrm {diam}}\nolimits X<M \end{array}}I_\gamma (T_{X}^k) \nonumber \\&\le 2^{M+2}C_1\sum _{j=n_k-R-m_k}^\infty I_\gamma \left( \frac{aj}{2v_a}\right) . \end{aligned}$$

(72)

The last line is independent of $s\in [0,1]$ and goes to 0 as $k\rightarrow \infty$.

Hence we have shown

$$\begin{aligned} \sup _{s\in [0,1]} \left\| \left[ \sum _{X\subset \Lambda _{n_k}} \int _{-\infty }^\infty dt W_\gamma (t) B(X,s,t,k),\; \Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right) \right] \right\| \rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$

(73)

The latter part of (67) can be estimated analogously. We divide the integral into $|t|\le S_{X}^k$ part and $|t|\ge S_X^{k}$ part. The $|t|\le S_{X}^k$ part can be treated as in (70) and we have

$$\begin{aligned}&\sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X \subset \Lambda _{n_k}{\setminus }\Lambda _{n_k-R} \end{array}} \int _{|t|\le S_{X}^k} dt \left| W_\gamma (t)\right| \left\| \left[ \tau _{t}^{\Phi (s)+\Psi _{k}(s),\Lambda _{n_k}} \left( \Psi _k'(X;s)\right) , \;\Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right) \right] \right\| \nonumber \\&\quad \le \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X \subset \Lambda _{n_k}{\setminus }\Lambda _{n_k-R} \end{array}} C_1C_{1a}\left\| W_\gamma \right\| _1e^{v_aS_X^k-ad(X,\Lambda _{L_k})} (2R)\sum _{y\in {{\mathbb {Z}}}} F(|y|)\left\| A\right\| \nonumber \\&\quad \le C_1C_{1a}\left\| W_\gamma \right\| _1 2^{2R}(2R)\sum _{y\in {{\mathbb {Z}}}} F(|y|) \sum _{l=(n_k-R-L_k)}^\infty e^{-\frac{al}{2}}\left\| A\right\| . \end{aligned}$$

(74)

The last line is independent of $s\in [0,1]$ and goes to 0 as $k\rightarrow \infty$. The $|t|\ge S_{X}^k$ part can be treated as in (71) and we have

$$\begin{aligned}&\sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X \subset \Lambda _{n_k}{\setminus }\Lambda _{n_k-R} \end{array}} \int _{|t|\ge S_{X}^k} dt \; \left| W_\gamma (t)\right| \left\| \tau _{t}^{\Phi (s)+\Psi _{k}(s),\Lambda _{n_k}} \left( \Psi _k'(X;s)\right) \right\| \nonumber \\&\quad \le \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X \subset \Lambda _{n_k}{\setminus }\Lambda _{n_k-R} \end{array}} 2C_1I_\gamma (S_X^k) \le 2^{2R+1}C_1\sum _{j=n_k-R-L_k}^\infty I_\gamma \left( \frac{aj}{2v_a}\right) . \end{aligned}$$

(75)

The last line is independent of $s\in [0,1]$ and goes to 0 as $k\rightarrow \infty$.

Hence we have shown

$$\begin{aligned} \sup _{s\in [0,1]}\left\| \left[ -D_k(s)+{\hat{D}}_{k,o}(s), \;\Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right) \right] \right\| \rightarrow 0. \end{aligned}$$

(76)

We also bound $-D_k(s)+{\hat{D}}_{k,o}(s)$ itself. From (59)

$$\begin{aligned}&\left\| -D_k(s)+{\hat{D}}_{k,o}(s)\right\| \nonumber \\&\quad \le \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ \mathop {\mathrm {diam}}\nolimits X<M \end{array}} \int _{|t|\le T_X^k} dt |W_\gamma (t)|C_1C_{3,a}\sum _{x\in X}e^{v_a|t|-a\cdot d(x,\left( \Lambda _{n_k-R}\right) ^c)} \nonumber \\&\qquad +2C_1\sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ \mathop {\mathrm {diam}}\nolimits X<M \end{array}}\int _{|t|\ge T_X^k} dt |W_\gamma (t)|\nonumber \\&\qquad +\sum _{X \subset \Lambda _{n_k}{\setminus }\Lambda _{n_k-R}} C_1\int _{-\infty }^\infty dt |W_\gamma (t)| \nonumber \\&\quad \le \left( C_1C_{3,a}M\left\| W_\gamma \right\| _1 \sum _{l=1}^\infty \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ \mathop {\mathrm {diam}}\nolimits X<M\\ d(X,\left( \Lambda _{n_k-R}\right) ^c)=l \end{array}} e^{-\frac{al}{2}} \right) \nonumber \\&\qquad +\left( 4C_1 \sum _{l=0}^\infty I_\gamma \left( \frac{al}{2v_a}\right) \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ \mathop {\mathrm {diam}}\nolimits X<M\\ d(X,\left( \Lambda _{n_k-R}\right) ^c)=l \end{array}} 1 \right) + \left( 2^{2R}C_1 \left\| W_\gamma \right\| _1\right) \nonumber \\&\quad \le \left( C_1C_{3,a}2^{M}M\left\| W_\gamma \right\| _1 \sum _{l=1}^\infty e^{-\frac{al}{2}} \right) +\left( 2^{M+3}C_1R \sum _{l=0}^\infty I_\gamma \left( \frac{al}{2v_a}\right) \right) +2^{2R}C_1 \left\| W_\gamma \right\| _1. \end{aligned}$$

(77)

In the second inequality, we used the fact that $T_X^k=0$ if $d(X,\left( \Lambda _{n_k-R}\right) ^c))=0$. The last line is finite and independent of $s\in [0,1]$ and $k\in {\mathbb {N}}$. Combining this with (65), we obtain

$$\begin{aligned}&\sup _{s\in [0,1]}\left\| \left[ -D_k(s)+{\hat{D}}_{k,o}(s),\; \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) - \Pi _{L_k}\left( \left( {\hat{\alpha }}_s^{(k,o)}\right) ^{-1}\left( A\right) \right) \right] \right\| \rightarrow 0, \nonumber \\&\qquad k\rightarrow \infty . \end{aligned}$$

(78)

From (76) and (78), we obtain (64).

From (64), we prove (62),

$$\begin{aligned}&\left\| \left( \alpha _s^{(k)}\right) ^{-1}\left( A\right) -\left( {\hat{\alpha }}_{s}^{(k,o)}\right) ^{-1}(A) \right\| \nonumber \\&\quad =\left\| A-\alpha _s^{(k)}\circ \left( {\hat{\alpha }}_{s}^{(k,o)}\right) ^{-1}(A) \right\| =\left\| \int _0^s du\; \frac{d}{du}\alpha _u^{(k)}\circ \left( {\hat{\alpha }}_{u}^{(k,o)}\right) ^{-1}(A) \right\| \nonumber \\&\quad =\left\| \int _0^s du\; \alpha _u^{(k)}\left( i\left[ -D_k(u)+{\hat{D}}_{k,o}(u),\; \left( {\hat{\alpha }}_{u}^{(k,o)}\right) ^{-1}(A) \right] \right) \right\| \le \varepsilon _k(A)\rightarrow 0, \nonumber \\&\quad \qquad k\rightarrow \infty , \end{aligned}$$

(79)

for any $l\in {\mathbb {N}}$ and $A\in {{\mathcal {A}}}_{\Lambda _l}$. Hence we have

$$\begin{aligned} \lim _{k\rightarrow \infty }\left\| \left( \alpha _s^{(k)}\right) ^{-1}\left( A\right) -\left( {\hat{\alpha }}_{s}^{(k,o)}\right) ^{-1}(A) \right\| =0, \end{aligned}$$

(80)

for any $A\in {{\mathcal {A}}}$. As we also have

$$\begin{aligned} \left\| \left( {\hat{\alpha }}_{s}^{(k,o)}\right) ^{-1}(A)-\left( \alpha _{s,o}\right) ^{-1}(A) \right\| \rightarrow 0,\quad k\rightarrow \infty , \end{aligned}$$

(81)

for any $A\in {{\mathcal {A}}}$ from [BMNS], we obtain

$$\begin{aligned} \left\| \left( \alpha _s^{(k)}\right) ^{-1}\left( A\right) -\left( \alpha _{s,o}\right) ^{-1}(A) \right\| \rightarrow 0,\quad k\rightarrow \infty , \end{aligned}$$

(82)

for any $A\in {{\mathcal {A}}}$. From this, we have

$$\begin{aligned} \left\| \alpha _{s,o}(A)-\alpha _s^{(k)}(A) \right\| = \left\| \alpha _s^{(k)}\left( \left( \alpha _s^{(k)}\right) ^{-1}- \left( \alpha _{s,o}\right) ^{-1} \right) \alpha _{s,o}(A) \right\| \rightarrow 0,\quad k\rightarrow \infty , \end{aligned}$$

(83)

for any $A\in {{\mathcal {A}}}$. Hence we have proven the Lemma.

Proof of Lemma 5.2

First we prove

$$\begin{aligned}&\sup _{s\in [0,1]}\left( \sum _{X\subset \Lambda _{m_k}}\int _{-\infty }^\infty dt\; \left| W_\gamma (t)\right| \left\| -\tau _t^{\Phi (s),\Lambda _{n_k}}\left( \Phi '(X;s)\right) +\tau _t^{\Phi (s)}\left( \Phi '(X;s)\right) \right\| \right) \rightarrow 0, \nonumber \\&\qquad k\rightarrow \infty . \end{aligned}$$

(84)

To prove this, for each $X\in {{\mathfrak {S}}}_{{{\mathbb {Z}}}}$ and $k\in {\mathbb {N}}$ we set

$$\begin{aligned} S_X^{(k)}:=\frac{a}{2v_a}d(\Lambda _{n_k}^c,X). \end{aligned}$$

(85)

With this $S_X^{(k)}$, we divide the integral into $|t|\le S_X^{(k)}$ part and $|t|\ge S_X^{(k)}$ part. By (61) and Definition 3.42., $|t|\le S_X^{(k)}$ part is bounded as

$$\begin{aligned}&\sum _{X\subset \Lambda _{m_k}}\int _{|t|\le S_X^{(k)}} dt\; \left| W_\gamma (t)\right| \left\| -\tau _t^{\Phi (s),\Lambda _{n_k}}\left( \Phi '(X;s)\right) +\tau _t^{\Phi (s)}\left( \Phi '(X;s)\right) \right\| \nonumber \\&\quad \le C_{4,a} e^{-\frac{a}{2}(n_k-m_k)}. \end{aligned}$$

(86)

Here $C_{4,a}$ is a positive constant which is independent of k, s. The right hand side is indepenednt of $s\in [0,1]$ and converges to 0 as $k\rightarrow \infty$. The $|t|\ge S_X^{(k)}$ part

$$\begin{aligned}&\sum _{X\subset \Lambda _{m_k}}\int _{|t|\ge S_X^{(k)}} dt\; \left| W_\gamma (t)\right| \left\| -\tau _t^{\Phi (s),\Lambda _{n_k}}\left( \Phi '(X;s)\right) +\tau _t^{\Phi (s)}\left( \Phi '(X;s)\right) \right\| \nonumber \\&\quad \le 2\sum _{X\subset \Lambda _{m_k}}\int _{|t|\ge S_X^{(k)}} dt\; \left| W_\gamma (t)\right| \left\| \Phi '(X;s) \right\| \le 4C_1\sum _{\begin{array}{c} X\subset \Lambda _{m_k}\\ \mathop {\mathrm {diam}}\nolimits (X)<M \end{array}}I_\gamma (S_X^{(k)})\nonumber \\&\quad =4C_1\sum _{l=n_k-m_k}^\infty \sum _{\begin{array}{c} X\subset \Lambda _{m_k}\\ \mathop {\mathrm {diam}}\nolimits (X)<M\\ d(X,\Lambda _{n_k}^c)=l \end{array} }I_\gamma (S_X^{(k)}) \le 4C_1\cdot 2^{M}\sum _{l=n_k-m_k}^\infty I_\gamma (\frac{a}{2v_a}l). \end{aligned}$$

(87)

Here, we used Definition 3.42. for the second inequality. In the third line, we recalled the definition of $S_X^{(k)}$ (85) and used the fact that for any finite set X in $\Lambda _{m_k}$ with $\mathop {\mathrm {diam}}\nolimits (X)<M$, the distance between X and $\Lambda _{n_k}^c$ is at least $n_k-m_k$. We also used the fact that for any $l\ge n_k-m_k$, the number of $X\subset \Lambda _{m_k}$ with $\mathop {\mathrm {diam}}\nolimits (X)<M$ such that $d(X,\Lambda _{n_k}^c)=l$ is at most $2^{M}$. The right hand side of (87) is independent of $s\in [0,1]$ goes to 0 as $k\rightarrow \infty$, because of (19). Hence we have shown (84). Similarly, we have

$$\begin{aligned}&\sup _{s\in [0,1]}\left( \sum _{X\subset \Lambda _{m_k}}\int _{-\infty }^\infty dt\; \left| W_\gamma (t)\right| \left\| -\tau _t^{{\tilde{\Phi }}(s),\Lambda _{n_k}}\left( {\tilde{\Phi }}'(X;s)\right) +\tau _t^{{\tilde{\Phi }}(s)}\left( {\tilde{\Phi }}'(X;s)\right) \right\| \right) \rightarrow 0, \nonumber \\&\qquad k\rightarrow \infty . \end{aligned}$$

(88)

Next we show

$$\begin{aligned}&\sup _{s\in [0,1]}\left( \int _{-\infty }^\infty dt\; \left| W_\gamma (t)\right| \sum _{\begin{array}{c} X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \end{array}} \left\| \tau _t^{{\tilde{\Phi }}(s)}\left( {\tilde{\Phi }}'(X;s)\right) -\tau _t^{ \Phi (s)}\left( \Phi '(X;s)\right) \right\| \right) \rightarrow 0, \nonumber \\&\qquad k\rightarrow \infty . \end{aligned}$$

(89)

To prove this, for each $X\in {{\mathfrak {S}}}_{{{\mathbb {Z}}}}$, we set

$$\begin{aligned}&R_X:=\min \left\{ d(X,Y)\mid Y\cap [0,\infty )\ne \emptyset ,\; Y\cap (-\infty ,-1]\ne \emptyset ,\;\mathop {\mathrm {diam}}\nolimits Y<M\right\} , \end{aligned}$$

(90)

$$\begin{aligned}&T_X:=\frac{a}{2v_a}R_X. \end{aligned}$$

(91)

With this $T_X$, we divide the integral into $|t|\le T_X$ part and $|t|\ge T_X$ part. We then have

$$\begin{aligned}&\sum _{\begin{array}{c} X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \end{array}} \int _{-\infty }^\infty dt\; \left| W_\gamma (t)\right| \left\| \tau _t^{{\tilde{\Phi }}(s)}\left( {\tilde{\Phi }}'(X;s)\right) -\tau _t^{ \Phi (s)}\left( \Phi '(X;s)\right) \right\| \end{aligned}$$

(92)

$$\begin{aligned}&\quad \le \sum _{\begin{array}{c} X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \end{array}} \int _{|t|\le T_X} dt\; \left| W_\gamma (t)\right| \left\| \tau _t^{{\tilde{\Phi }}(s)}\left( \Phi '(X;s)\right) -\tau _t^{ \Phi (s)}\left( \Phi '(X;s)\right) \right\| \end{aligned}$$

(93)

$$\begin{aligned}&\qquad + \sum _{\begin{array}{c} X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \end{array}} \int _{|t|\le T_X} dt\; \left| W_\gamma (t)\right| \left\| \tau _t^{{\tilde{\Phi }}(s)}\left( {\tilde{\Phi }}'(X;s)-\Phi '(X;s)\right) \right\| \end{aligned}$$

(94)

$$\begin{aligned}&\qquad + \sum _{\begin{array}{c} X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \end{array}} \int _{|t|\ge T_X} dt\; \left| W_\gamma (t)\right| \left\| \tau _t^{{\tilde{\Phi }}(s)}\left( {\tilde{\Phi }}'(X;s)\right) -\tau _t^{ \Phi (s)}\left( \Phi '(X;s)\right) \right\| . \end{aligned}$$

(95)

The first part (93) is bounded by use of the (60) as

$$\begin{aligned} |(93)| \le \left\| W_\gamma \right\| _1 C_1C_{2,a}M \sum _{l=m_k-M}^\infty \sum _{\begin{array}{c} X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \\ \mathop {\mathrm {diam}}\nolimits X<M\\ d(X,\{0\})=l \end{array}} e^{\frac{a}{2} (-l+M)} \le C_{5,a}\sum _{l=m_k-M}^\infty e^{-\frac{al}{2}}. \end{aligned}$$

In the last line, we used $R_X\le d(X,\{0,-1\})\le d(\{x\},\{0\})$ for all $x\in X$ and $d(X,\{0\})-M\le d(X,[-M,M])\le R_X$. (Recall we assumed $M>2$ in the beginning of this section.) We also used the fact that the number of X with $\mathop {\mathrm {diam}}\nolimits X<M$ and $d(X,\{0\})=l$ is bounded by $2^{M}$, and introduced a new constant $C_{5,a}:=2^{M}M \left\| W_\gamma \right\| _1 C_1C_{2,a}e^{\frac{a}{2}{M}}$. The right hand side is independent of $s\in [0,1]$ and goes to 0 as $k\rightarrow \infty$. The second term (94) is 0 for k large enough. The third term (95) can be evaluated as in (87). We have for $m_k>2M$,

$$\begin{aligned} |(95)|&\le 4C_1 \sum _{\begin{array}{c} X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \\ \mathop {\mathrm {diam}}\nolimits X<M \end{array}} I_\gamma (T_X) \le 4C_1 \sum _{l=m_k-M}^\infty \sum _{\begin{array}{c} X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \\ \mathop {\mathrm {diam}}\nolimits X<M\\ d(X,\{0\})=l \end{array}}I_\gamma \left( \frac{a}{2v_a}(l-M)\right) \nonumber \\&\le 4C_1 2^{M} \sum _{l=m_k-2M}^\infty I_\gamma \left( \frac{a}{2v_a}l\right) . \end{aligned}$$

(96)

Here we used $d(X,\{0\})-M\le R_X$, for the second inequality.

The right hand side is independent of $s\in [0,1]$ and goes to 0 as $k\rightarrow \infty$. Hence we have shown (89). Similarly, we obtain

$$\begin{aligned}&\sup _{s\in [0,1]}\left( \int _{-\infty }^\infty dt\; \left| W_\gamma (t)\right| \sum _{\begin{array}{c} X\subset \Lambda _{n_k}\\ X\cap \Lambda _{m_k}^c\ne \emptyset \end{array}} \left\| \tau _t^{{\tilde{\Phi }}(s),\Lambda _{n_k}}\left( {\tilde{\Phi }}'(X,s)\right) -\tau _t^{\Phi (s),\Lambda _{n_k}}\left( \Phi '(X,s)\right) \right\| \right) \rightarrow 0, \nonumber \\&\qquad k\rightarrow \infty . \end{aligned}$$

(97)

From (89), we have

$$\begin{aligned} \int _{-\infty }^\infty dt\; \left| W_\gamma (t)\right| \;\; \left( \sum _{X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}} \left\| \tau _t^{{\tilde{\Phi }}(s)} \left( {\tilde{\Phi }}'(X;s)\right) - \tau _t^{\Phi (s)} \left( \Phi '(X;s)\right) \right\| \right) <\infty . \end{aligned}$$

(98)

Therefore, we may define

$$\begin{aligned} V(s):= \int _{-\infty }^\infty dt\; W_\gamma (t)\;\; \left( \sum _{X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}} \tau _t^{{\tilde{\Phi }}(s)} \left( {\tilde{\Phi }}'(X;s)\right) - \tau _t^{\Phi (s)} \left( \Phi '(X;s)\right) \right) \in {{\mathcal {A}}}, \end{aligned}$$

(99)

and from (84), (88), (89), (97), we obtain (27).$\quad \square$

On-site Group Symmetry

For a Hilbert space ${{\mathcal {K}}}$, we denote by ${{\mathcal {U}}}({{\mathcal {K}}})$ the set of all unitaries on ${{\mathcal {K}}}$. Let G be a finite group and $w:G\rightarrow {{\mathcal {U}}}({{\mathbb {C}}}^{2S+1})$ a unitary representation of G on ${{\mathbb {C}}}^{2S+1}$. Then there is an action $T:G\rightarrow {\mathrm {Aut}}{{\mathcal {A}}}$ of G on ${{\mathcal {A}}}$ such that

$$\begin{aligned} T_g\left( A \right) =\left( \bigotimes _I w(g)\right) A \left( \bigotimes _I w(g)^*\right) ,\quad g\in G,\quad A\in {{\mathcal {A}}}_I, \end{aligned}$$

(100)

for any finite interval I of ${{\mathbb {Z}}}$. A state $\varphi$ on ${{\mathcal {A}}}$ is G-invariant if $\varphi \circ T_g=\varphi$ for any $g\in G$. As $T_g ({{\mathcal {A}}}_R)={{\mathcal {A}}}_R$, the restriction $T_{g,R}:=\left. T_g\right| _{{{\mathcal {A}}}_R}$ is a $*$-automorphism on ${{\mathcal {A}}}_R$.

In [M2], Matsui introduced the projective representation of G associated to pure split G-invariant states. As in Theorem 2.2, it is unique up to unitary conjugacy and a phase, and the cohomology class is independent of the choice of the projective representation.

Theorem B.1

Let $\varphi$ be a G-invariant pure state on ${{\mathcal {A}}}$, which satisfies the split property. Let $\varphi _R$ be the restriciton of $\varphi$ to ${{\mathcal {A}}}_R$, and $({{\mathcal {H}}}_{\varphi _R},\pi _{\varphi _R},\Omega _{\varphi _R})$ be the GNS triple of $\varphi _R$. Then there are a Hilbert space ${{\mathcal {K}}}_\varphi$, a $*$-isomorphism $\iota _\varphi : \pi _{\varphi _R}\left( {{\mathcal {A}}}_R\right) {''}\rightarrow B({{\mathcal {K}}}_{\varphi })$, and a projective unitary representation $U_\varphi :G\rightarrow {{\mathcal {U}}}({{\mathcal {K}}}_\varphi )$ on ${{\mathcal {K}}}_{\varphi }$ such that

$$\begin{aligned} \iota _\varphi \circ \pi _{\varphi _R}\circ T_{g,R}\left( A\right) =U_\varphi (g) \left( \iota _\varphi \circ \pi _{\varphi _R}\left( A\right) \right) U_\varphi (g)^*,\quad A\in {{\mathcal {A}}}_R,\quad g\in G. \end{aligned}$$

These ${{\mathcal {K}}}_\varphi$, $\iota _\varphi$, $U_{\varphi }$ are unique in the following sense.: If a Hilbert space ${\tilde{{{\mathcal {K}}}}}_\varphi$, a $*$-isomorphism ${\tilde{\iota }}_\varphi : \pi _{\varphi _R}\left( {{\mathcal {A}}}_R\right) {''}\rightarrow B({\tilde{{{\mathcal {K}}}}}_{\varphi })$, and a projective unitary representation ${\tilde{U}}_\varphi :G\rightarrow {{\mathcal {U}}}({\tilde{{{\mathcal {K}}}}}_\varphi )$ on ${\tilde{{{\mathcal {K}}}}}_{\varphi }$ satisfy

$$\begin{aligned} {\tilde{\iota }}_\varphi \circ \pi _{\varphi _R}\circ T_{g,R}\left( A\right) ={\tilde{U}}_\varphi (g) \left( {\tilde{\iota }}_\varphi \circ \pi _{\varphi _R}\left( A\right) \right) {{\tilde{U}}_\varphi (g)}^*,\quad A\in {{\mathcal {A}}}_R,\quad g\in G, \end{aligned}$$

then there is a unitary $W:{{\mathcal {K}}}_\varphi \rightarrow {\tilde{{{\mathcal {K}}}}}_\varphi$ and $c: G\rightarrow {{\mathbb {T}}}$ such that

$$\begin{aligned}&W\left( \iota _\varphi \left( x\right) \right) W^*= {\tilde{\iota }}_\varphi \left( x\right) , \quad x\in \pi _{\varphi _R}\left( {{\mathcal {A}}}_R\right) {''},\\&c(g) WU_\varphi (g) W^*={\tilde{U}}_\varphi (g),\quad g\in G. \end{aligned}$$

In particular, the cohomology class of $U_\varphi$ is equal to that of ${\tilde{U}}_{\varphi }$.

The same argument as the proof of Theorem 2.6 shows that the cohomology class is an invariant of factorizable automorphic equivalence, preserving G-symmetry.

Theorem B.2

Let $\varphi _1,\varphi _2$ be G-invariant pure states satisfying the split property. Suppose that there exists an automorphism $\alpha$ on ${{\mathcal {A}}}$ such that

$$\begin{aligned} \varphi _2=\varphi _1\circ \alpha \quad \text {and}\quad \alpha \circ T_g=T_g\circ \alpha ,\quad g\in G. \end{aligned}$$

(101)

Furthermore, assume that there are automorphisms $\alpha _R$, $\alpha _L$ on ${{\mathcal {A}}}_R$, ${{\mathcal {A}}}_L$ respectively, and a unitary W in ${{\mathcal {A}}}$ such that

$$\begin{aligned} \alpha _R\circ T_{g,R}=T_{g,R}\circ \alpha _R,\quad g\in G \end{aligned}$$

(102)

and

$$\begin{aligned} \alpha \circ \left( \alpha _L^{-1}\otimes \alpha _R^{-1}\right) (A)= WAW^*,\quad A\in {{\mathcal {A}}}. \end{aligned}$$

Then the the cohomology class of the associated projective representations of $\varphi _1$ and $\varphi _2$ are equal.

From this, we can show that the cohomology class is invariant of $C^1$-classification.

Theorem B.3

Let $\Phi :[0,1]\ni s \rightarrow \Phi (s):=\{\Phi (X;s)\}_{X\in {{\mathfrak {S}}}_{{\mathbb {Z}}}}\in {{{\mathcal {B}}}_f}$ be a $C^1$-path of interactions, satisfying the Condition B with

6’. For each $s\in [0,1]$, $\Phi (s)$ is G-invariant i.e.,

$$\begin{aligned} T_g\left( \Phi (X;s)\right) =\Phi (X;s),\quad g\in G,\quad X\in {\mathfrak {S}}_{{\mathbb {Z}}}, \end{aligned}$$

instead of 6. Then the cohomology class of the associated representation of the ground state does not change along the path.