An Invariant of Symmetry Protected Topological Phases with On-Site Finite Group Symmetry for Two-Dimensional Fermion Systems

Abstract

We consider SPT-phases with on-site finite group G symmetry for two-dimensional Fermion systems. We derive an invariant of the classification.

Appendices

Basic Notations

For a Hilbert space ${\mathcal {H}}$, $B({\mathcal {H}})$ denotes the set of all bounded operators on ${\mathcal {H}}$, while ${\mathcal {U}}({\mathcal {H}})$ denotes the set of all unitaries on ${\mathcal {H}}$. If $V:{\mathcal {H}}_1\rightarrow {\mathcal {H}}_2$ is a linear map from a Hilbert space ${\mathcal {H}}_1$ to another Hilbert space ${\mathcal {H}}_2$, then ${\mathrm {Ad}}(V):B({\mathcal {H}}_1)\rightarrow B({\mathcal {H}}_2)$ denotes the map ${\mathrm {Ad}}(V)(x):=V x V^*$, $x\in B({\mathcal {H}}_1)$. Occasionally we write ${\mathrm {Ad}}_V$ instead of ${\mathrm {Ad}}(V)$. For a $C^{*}$-algebra ${\mathcal {B}}$ and $v\in {\mathcal {B}}$, we set ${\mathrm {Ad}}(v)(x):={\mathrm {Ad}}_{v}(x):=vxv^{*}$, $x\in {\mathcal {B}}$.

For a state $\omega$ on a $C^*$-algebra ${\mathfrak {B}}$, we denote by $({\mathcal {H}}_\omega , \pi _\omega ,\Omega _\omega )$ its GNS triple. For a $C^*$-algebra ${\mathfrak {B}}$, we denote by ${\mathcal {U}}({\mathfrak {B}})$ the set of all unitaries in ${\mathfrak {B}}$. For a $C^*$-algebra ${\mathfrak {B}}$, ${\mathfrak {B}}_{+,1}$ denotes the set of all positive elements of ${\mathfrak {B}}$ with norm less than or equal to 1. For states $\omega ,\varphi$ on a $C^*$-algebra ${\mathfrak {B}}$, we write $\omega \simeq \varphi$ if they are equivalent and $\omega \sim _{{q.e.}}\varphi$ if they are quasi-equivalent. We denote by ${\mathrm {Aut}}{\mathcal {B}}$ the group of automorphisms on a $C^{*}$-algebra ${\mathcal {B}}$. The group of inner automorphisms on a unital $C^{*}$-algebra ${\mathcal {B}}$ is denoted by ${\mathrm {Inn}}{\mathcal {B}}$. For $\gamma _1,\gamma _2\in {\mathrm {Aut}}({\mathcal {B}})$, $\gamma _1=({\mathrm{inner}})\circ \gamma _2$ means there is some unitary u in ${\mathcal {B}}$ such that $\gamma _1={\mathrm {Ad}}(u)\circ \gamma _2$. For a unital $C^{*}$-algebra ${\mathcal {B}}$, the unit of ${\mathcal {B}}$ is denoted by $\mathbb {I}_{{\mathcal {B}}}$. For a Hilbert space we write $\mathbb {I}_{{\mathcal {H}}}:=\mathbb {I}_{{\mathcal {B}}({\mathcal {H}})}$.But we occationally omit ${\mathcal {B}},{\mathcal {H}}$ etc. For a state $\varphi$ on ${\mathcal {B}}$ and a $C^{*}$-subalgebra ${\mathcal {C}}$ of ${\mathcal {B}}$, $\varphi \vert _{{\mathcal {C}}}$ indicates the restriction of $\varphi$ to ${\mathcal {C}}$.

The center of a von Neumann algebra ${\mathcal {M}}$ is denoted by $Z({\mathcal {M}})$.

To denote the composition of automorphisms $\alpha _1$, $\alpha _2$, all of $\alpha _1\circ \alpha _2$, $\alpha _1\alpha _2$, $\alpha _1\cdot \alpha _2$ are used. Frequently, the first one serves as a bracket to visually separate a group of operators.

Graded von Neumann Algebras

In this section we collect facts we use about graded von Neumann algebras. See [BO] for further explanation. A graded von Neumann algebra is a pair $({\mathcal {M}},\theta )$ with ${\mathcal {M}}$ a von Neumann algebra $\theta$ an involutive automorphism on ${\mathcal {M}}$, $\theta ^2 = \mathrm {Id}$. The even/odd part of ${\mathcal {M}}$ with respect to the grading is denoted by ${\mathcal {M}}^{(0)}$/${\mathcal {M}}^{(1)}$. If ${\mathcal {M}}\subset {\mathcal {B}}({\mathcal {H}})$ and there is a self-adjoint unitary $\Gamma$ on ${\mathcal {H}}$ such that $\mathrm {Ad}_{\Gamma }|_{\mathcal {M}}= \theta$, then we call $({\mathcal {M}},\theta )$ a spatially graded von Neumann algebra with grading operator $\Gamma$. We say $({\mathcal {M}},\theta )$ is balanced if ${\mathcal {M}}$ contains an odd self-adjoint unitary. If $Z({\mathcal {M}})\cap {\mathcal {M}}^{(0)}={\mathbb {C}}\mathbb {I}$ for the center $Z({\mathcal {M}})$ of ${\mathcal {M}}$, we say $({\mathcal {M}}, \theta )$ is central.

For a homogeneous state $\omega$ on a graded $C^*$-algebra ${\mathfrak {B}}$, there is a self-adjoint unitary $\Gamma _\omega$ implementing the grading with respect to $\pi _\omega$. As a result, the grading extends to the von Neumann algebra $\pi _\omega ({\mathfrak {B}})''$ by ${\mathrm {Ad}}\Gamma _\omega$. We always consider this extension without mentioning explicitly.

Lemma B.1

Let $({\mathcal {M}},\Gamma )$ be a balanced central graded von Neumann algebra on a Hilbert space ${\mathcal {H}}$. Then both of ${\mathcal {M}}$ and ${\mathcal {M}}^{(0)}$ are either a factor or a direct sum of two factors of the same type.

Proof

The proof is basically the same as part of Proposition 2.9 of [BO]. From Lemma A.2 of [BO], if ${\mathcal {M}}$ is not a factor, it is a direct sum of two factors of the same type which maps to each other by ${\mathrm {Ad}}\Gamma$.

Let U be a self-adjoint odd unitary in ${\mathcal {M}}$. Suppose that ${\mathcal {M}}^{(0)}$ is not a factor. Then there exists a projection z in $Z({\mathcal {M}}^{(0)})$ which is not 0 nor $\mathbb {I}$. For such a projection, we have $z+{\mathrm {Ad}}_{U}(z)\in {\mathcal {M}}^{(0)}\cap \left( {{\mathcal {M}}^{(0)}}\right) '\cap \{U\}'=Z({\mathcal {M}}) \cap {\mathcal {M}}^{(0)}={\mathbb {C}}\mathbb {I}$, which then implies that $z+{\mathrm {Ad}}_{U}(z)=\mathbb {I}$. (We note that for orthogonal projections p, q satisfying $p+q=t\mathbb {I}$ with $t\in {\mathbb {R}}$, either $p+q=\mathbb {I}$ or $p=0,\,\mathbb {I}$ holds, by considering the spectrum of $p=t\mathbb {I}-q$.)

We claim $Z({\mathcal {M}}^{(0)})={\mathbb {C}}z+{\mathbb {C}}\mathbb {I}$. Now, for any projection s in $Z({\mathcal {M}}^{(0)})$, zs is a projection in $Z({\mathcal {M}}^{(0)})$. Therefore either $zs=0$ or $zs+{\mathrm {Ad}}_{U}(zs)=\mathbb {I}$. The latter is possible only if $zs=z$ because $z+{\mathrm {Ad}}_{U}(z)=\mathbb {I}$. Similarly, we have $(\mathbb {I}-z)s=0$ or $(\mathbb {I}-z)s=\mathbb {I}-z$. Hence we have $Z({\mathcal {M}}^{(0)})={\mathbb {C}}z+{\mathbb {C}}\mathbb {I}$, proving the claim.

Hence ${\mathcal {M}}^{(0)}$ is a summation of two factors ${\mathcal {M}}^{(0)}={\mathcal {M}}^{(0)}z\oplus {\mathcal {M}}^{(0)}(\mathbb {I}-z)$. Because ${\mathrm {Ad}}U(z)=\mathbb {I}-z$, these factors ${\mathcal {M}}^{(0)}z$, ${\mathcal {M}}^{(0)}(\mathbb {I}-z)$ are isomorphic. In particular, they have the same type. $\square$

Let $({\mathcal {M}}_1,{\mathrm {Ad}}_{\Gamma _1})$ and $({\mathcal {M}}_2,{\mathrm {Ad}}_{\Gamma _2})$ be spatially graded von Neumann algebras acting on on ${\mathcal {H}}_1$, ${\mathcal {H}}_2$ with grading operators $\Gamma _1$, $\Gamma _2$. We define a product and involution on the algebraic tensor product ${\mathcal {M}}_1\odot {\mathcal {M}}_2$ by

$$\begin{aligned} (a_1{\hat{\otimes }} b_1)(a_2{\hat{\otimes }} b_2)&=(-1)^{\partial b_1\partial a_2}(a_1a_2{\hat{\otimes }} b_1b_2), \nonumber \\ (a{\hat{\otimes }} b)^*&=(-1)^{\partial a\partial b} a^*{\hat{\otimes }} b^*. \end{aligned}$$

(B.1)

for homogeneous elementary tensors. The algebraic tensor product with this multiplication and involution is a $*$-algebra, denoted ${\mathcal {M}}_1\,\hat{\odot }\,{\mathcal {M}}_2$. On the Hilbert space ${\mathcal {H}}_1\otimes {\mathcal {H}}_2$,

$$\begin{aligned} \pi (a{\hat{\otimes }} b) :=a\Gamma _1^{\partial b}\otimes b \end{aligned}$$

(B.2)

for homogeneous $a\in {\mathcal {M}}_1$, $b\in {\mathcal {M}}_2$ defines a faithful $*$-representation of ${\mathcal {M}}_1\,\hat{\odot }\,{\mathcal {M}}_2$.

We call the von Neumann algebra generated by $\pi ({\mathcal {M}}_1\,\hat{\odot }\,{\mathcal {M}}_2)$ the graded tensor product of $({\mathcal {M}}_1,{\mathcal {H}}_1,\Gamma _1)$ and $({\mathcal {M}}_2,{\mathcal {H}}_2,\Gamma _2)$ and denote it by ${\mathcal {M}}_1 {\hat{\otimes }} {\mathcal {M}}_2$. It is simple to check that ${\mathcal {M}}_1 {\hat{\otimes }} {\mathcal {M}}_2$ is a spatially graded von Neumann algebra with a grading operator $\Gamma _1\otimes \Gamma _2$.

For $a\in {\mathcal {M}}_1$ and homogeneous $b\in {\mathcal {M}}_2$, we denote $\pi (a{\hat{\otimes }} b)$ by $a{{\hat{\otimes }}} b$, embedding ${\mathcal {M}}_1\,\hat{\odot }\,{\mathcal {M}}_2$ in ${\mathcal {M}}_1{\hat{\otimes }}{\mathcal {M}}_2$. Note that $\partial (a{\hat{\otimes }} b) = \partial (a) + \partial (b)$ for homogeneous $a\in {\mathcal {M}}_1$ and $b\in {\mathcal {M}}_2$.

Lemma B.2

For each $i=1,2$, let $({\mathcal {M}}_i,\Gamma _i)$ be balanced central graded von Neumann algebras on a Hilbert space ${\mathcal {H}}_i$. Suppose that ${\mathcal {M}}_1{\hat{\otimes }} {\mathcal {M}}_2$ is of type I factor. Then both of ${\mathcal {M}}_1$ and ${\mathcal {M}}_2$ are type I.

Proof

By Lemma B.1, all of ${\mathcal {M}}_1$, ${\mathcal {M}}_2$, ${\mathcal {M}}_1^{(0)}$, ${\mathcal {M}}_2^{(0)}$ are either a factor or a direct sum of two factors of the same type. Then by Lemma A.1 of [BO], the type of ${\mathcal {M}}_i$ and ${\mathcal {M}}_i^{(0)}$ are the same for each $i=1,2$.

Because ${\mathcal {M}}_1{\hat{\otimes }} {\mathcal {M}}_2$ is a type I factor, it has a faithful semifinite normal trace $\tau$ whose restriction to ${\mathcal {M}}_1^{(0)}\otimes {\mathcal {M}}_2^{(0)}$ is also a faithful semifinite normal trace. Therefore, from Theorem 2.15 of [T], ${\mathcal {M}}_1^{(0)}\otimes {\mathcal {M}}_2^{(0)}$ is semifinite. From Theorem 2.30, it means both of ${\mathcal {M}}_1^{(0)}$ and ${\mathcal {M}}_2^{(0)}$ are semifinite. Because ${\mathcal {M}}_1{\hat{\otimes }} {\mathcal {M}}_2$ is a type I factor, for the set of orthogonal projections ${\mathcal {P}}\left( {\mathcal {M}}_1{\hat{\otimes }} {\mathcal {M}}_2\right)$ of ${\mathcal {M}}_1{\hat{\otimes }} {\mathcal {M}}_2$, $\tau \left( {\mathcal {P}}\left( {\mathcal {M}}_1{\hat{\otimes }} {\mathcal {M}}_2\right) \right)$ is a countable set. It means that for the set of orthogonal projections ${\mathcal {P}}\left( {\mathcal {M}}_1^{(0)}\otimes {\mathcal {M}}_2^{(0)}\right)$ of ${\mathcal {M}}_1^{(0)}\otimes {\mathcal {M}}_2^{(0)}$, $\tau \left( {\mathcal {P}}\left( {\mathcal {M}}_1^{(0)}{\hat{\otimes }} {\mathcal {M}}_2^{(0)}\right) \right)$ is also countable. It means that ${\mathcal {M}}_1^{(0)}\otimes {\mathcal {M}}_2^{(0)}$ is not of type II. It means both of ${\mathcal {M}}_1^{(0)}$ and ${\mathcal {M}}_2^{(0)}$ are type I, hence both of ${\mathcal {M}}_1$ and ${\mathcal {M}}_2$ are type I. $\square$

Lemma B.3

Let ${\mathcal {H}}$ be a Hilbert space with a self-adjoint unitary $\Gamma$ that gives a grading for ${\mathcal {B}}({\mathcal {H}})$. Let ${\mathcal {M}}_1$, ${\mathcal {M}}_2$ be ${\mathrm {Ad}}_{\Gamma }$-invariant type I von Neumann subalgebras of ${\mathcal {B}}({\mathcal {H}})$ with ${\mathcal {M}}_1 \vee {\mathcal {M}}_2 = {\mathcal {B}}({\mathcal {H}})$. Suppose with respect to the grading given by ${\mathrm {Ad}}_{\Gamma }$ that both of ${\mathcal {M}}_1$ and ${\mathcal {M}}_2$ have a center of the form $Z\left( {\mathcal {M}}_i\right) ={\mathbb {C}}\mathbb {I}+{\mathbb {C}}V_i$ with a self-adjoint odd unitary $V_i$. Suppose further that

$$\begin{aligned} ab- (-1)^{\partial a \partial b} b a = 0,\quad \text {for homogeneous}\quad a\in {\mathcal {M}}_1, b \in {\mathcal {M}}_2. \end{aligned}$$

(B.3)

Then there are Hilbert spaces ${\mathcal {K}}_1,{\mathcal {K}}_2$ and a unitary $W:{\mathcal {H}}\rightarrow {\mathcal {K}}_1\otimes {\mathcal {K}}_2\otimes {\mathbb {C}}^2$ such that

$$\begin{aligned} \begin{aligned} {\mathrm {Ad}}W \left( {\mathcal {M}}_1^{(0)}\right)&={\mathcal {B}}({\mathcal {K}}_1)\otimes {\mathbb {C}}\mathbb {I}_{{\mathcal {K}}_2}\otimes {\mathbb {C}}\mathbb {I}_{{\mathbb {C}}^2},\\ {\mathrm {Ad}}W \left( {\mathcal {M}}_2^{(0)}\right)&={\mathbb {C}}\mathbb {I}_{{\mathcal {K}}_1} \otimes {\mathcal {B}}({\mathcal {K}}_2)\otimes {\mathbb {C}}\mathbb {I}_{{\mathbb {C}}^2},\\ {\mathrm {Ad}}W (V_1)&=\mathbb {I}_{{\mathcal {K}}_1}\otimes \mathbb {I}_{{\mathcal {K}}_2}\otimes \sigma _z,\\ {\mathrm {Ad}}W(V_2)&=\mathbb {I}_{{\mathcal {K}}_1}\otimes \mathbb {I}_{{\mathcal {K}}_2}\otimes \sigma _x. \end{aligned} \end{aligned}$$

(B.4)

Proof

By Proposition 2.9 of [BO] (with G a trivial group), ${\mathcal {M}}_i^{(0)}$ $i=1,2$ are type I factors. By the assumption (B.3), ${\mathcal {M}}_1^{(0)}$ and ${\mathcal {M}}_2^{(0)}$ commute. From (B.3) and the fact that $V_i$ belongs to center of ${\mathcal {M}}_i$, we know that both of $V_1$ and $V_2$ commute with ${\mathcal {M}}_1^{(0)}$ and ${\mathcal {M}}_2^{(0)}$. As a result, there are Hilbert spaces ${\mathcal {K}}_1,{\mathcal {K}}_2,{\mathcal {K}}_3$ and a unitary ${\hat{W}}: {\mathcal {H}}\rightarrow {\mathcal {K}}_1\otimes {\mathcal {K}}_2\otimes {\mathcal {K}}_3$ such that

$$\begin{aligned} \begin{aligned} {\mathrm {Ad}}{\hat{W}} \left( {\mathcal {M}}_1^{(0)}\right)&={\mathcal {B}}({\mathcal {K}}_1)\otimes {\mathbb {C}}\mathbb {I}_{{\mathcal {K}}_2}\otimes {\mathbb {C}}\mathbb {I}_{{\mathcal {K}}_3},\\ {\mathrm {Ad}}{\hat{W}} \left( {\mathcal {M}}_2^{(0)}\right)&={\mathbb {C}}\mathbb {I}_{{\mathcal {K}}_1} \otimes {\mathcal {B}}({\mathcal {K}}_2)\otimes {\mathbb {C}}\mathbb {I}_{{\mathcal {K}}_3},\\ {\mathrm {Ad}}{\hat{W}} (V_1)&=\mathbb {I}_{{\mathcal {K}}_1}\otimes \mathbb {I}_{{\mathcal {K}}_2}\otimes y_1,\\ {\mathrm {Ad}}{\hat{W}}(V_2)&=\mathbb {I}_{{\mathcal {K}}_1}\otimes \mathbb {I}_{{\mathcal {K}}_2}\otimes y_2, \end{aligned} \end{aligned}$$

(B.5)

with $y_1,y_2$ self-adjoint unitaries on ${\mathcal {K}}_3$. Because $V_2 V_1 V_2^*=-V_1$, we have $y_2 y_1y_2^*=-y_1$. With $y_1=r_{1+}-r_{1-}$ as a spectral projection, this means that $y_2 r_{1\pm }y_2=r_{1\mp }$, and $u:=r_{1+} y_2 r_{1-}: r_{1-}{\mathcal {K}}_3\rightarrow r_{1+}{\mathcal {K}}_3$ is a unitary. Let $\{e_1,e_2\}$ be the standard basis of ${\mathbb {C}}^2$ and set $v:{\mathcal {K}}_3\rightarrow {\mathbb {C}}^2\otimes r_{1+}{\mathcal {K}}$ be a unitary given by

$$\begin{aligned} v\xi := e_1\otimes r_{1+}\xi +e_2\otimes ur_{1-}\xi ,\quad \xi \in {\mathcal {K}}_3. \end{aligned}$$

(B.6)

It is then straightforward to show that

$$\begin{aligned} \begin{aligned} {\mathrm {Ad}}v(y_1)=\sigma _z\otimes \mathbb {I}_{r_{1+}{\mathcal {K}}},\\ {\mathrm {Ad}}v(y_2)=\sigma _x\otimes \mathbb {I}_{r_{1+}{\mathcal {K}}}. \end{aligned} \end{aligned}$$

(B.7)

From this, (B.5) and ${\mathcal {M}}_1 \vee {\mathcal {M}}_2 = {\mathcal {B}}({\mathcal {H}})$, we see that $r_{1+}{\mathcal {K}}$ is one-dimensional. Hence $W:=(\mathbb {I}_{{\mathcal {K}}_1}\otimes \mathbb {I}_{{\mathcal {K}}_2}\otimes v){\hat{W}} : {\mathcal {H}}\rightarrow {\mathcal {K}}_1\otimes {\mathcal {K}}_2\otimes {\mathbb {C}}^2$ defines a unitary satisfying (B.4). $\square$

Miscellaneous Lemmas

It is elementary to show the following Lemma. We omit the proof.

Lemma C.1

For $\sigma =L,R$, let ${\mathcal {B}}_\sigma$ be a graded $C^*$-algebra with a grading automorphism $\Theta _\sigma$. Let $({\mathcal {H}}_\sigma ,\pi _\sigma )$ be an irreducible representation of ${\mathcal {B}}_\sigma$ with a self-adjoint unitary $\Gamma _\sigma$ implementing $\Theta _\sigma$. Let $\zeta _\sigma \in {\mathrm {Aut}}^{(0)}({\mathcal {B}}_\sigma )$. Then the followings hold.

(i):

If each $\zeta _\sigma$ is implemented by a unitary $u_\sigma$ on $({\mathcal {H}}_\sigma ,\pi _\sigma )$, i.e., ${\mathrm {Ad}}\left( u_\sigma \right) \circ \pi _\sigma =\pi _\sigma \circ \zeta _\sigma$, then $u_\sigma$ is homogeneous with respect to ${\mathrm {Ad}}(\Gamma _\sigma )$ and

$$\begin{aligned} \begin{aligned}&{\mathrm {Ad}}\left( \mathbb {I}\otimes u_R\right) \circ \left( \pi _L{\hat{\otimes }} \pi _R\right) =\left( \pi _L{\hat{\otimes }} \pi _R\right) \circ \left( {\mathrm {id}}_{{\mathcal {B}}_L}{\hat{\otimes }} \zeta _R\right) ,\\&{\mathrm {Ad}}\left( u_L\otimes \Gamma _R^{\partial u_L}\right) \circ \left( \pi _L{\hat{\otimes }} \pi _R\right) =\left( \pi _L{\hat{\otimes }} \pi _R\right) \left( \zeta _L{\hat{\otimes }} {\mathrm {id}}_{{\mathcal {B}}_R}\right) . \end{aligned} \end{aligned}$$

(C.1)

Here $\partial u_L$ denotes the grade of $u_L$ with respect to ${\mathrm {Ad}}(\Gamma _L)$.

(ii):

Suppose that there are unitaries $U_\sigma$, $\sigma =L,R$ on ${\mathcal {H}}_L\otimes {\mathcal {H}}_R$ such that

$$\begin{aligned} \begin{aligned}&{\mathrm {Ad}}\left( U_R\right) \circ \left( \pi _L{\hat{\otimes }} \pi _R\right) =\left( \pi _L{\hat{\otimes }} \pi _R\right) \circ \left( {\mathrm {id}}_{{\mathcal {B}}_L}{\hat{\otimes }} \zeta _R\right) ,\\&{\mathrm {Ad}}\left( U_L\right) \circ \left( \pi _L{\hat{\otimes }} \pi _R\right) =\left( \pi _L{\hat{\otimes }} \pi _R\right) \left( \zeta _L{\hat{\otimes }} {\mathrm {id}}_{{\mathcal {B}}_R}\right) . \end{aligned} \end{aligned}$$

(C.2)

Then there are unitaries $u_\sigma \in {\mathcal {U}}({\mathcal {H}}_\sigma )$ such that ${\mathrm {Ad}}\left( u_\sigma \right) \circ \pi _\sigma =\pi _\sigma \circ \zeta _\sigma$, and

$$\begin{aligned} \mathbb {I}\otimes u_R=U_R,\quad u_L\otimes \Gamma _R^{\partial u_L}=U_L. \end{aligned}$$

(C.3)