Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries

We propose the definitions of many-body topological invariants to detect symmetry-protected topological phases protected by point group symmetry, using partial point group transformations on a given short-range entangled quantum ground state. Here, partial point group transformations $g_D$ are defined by point group transformations restricted to a spatial subregion $D$, which is closed under the point group transformations and sufficiently larger than the bulk correlation length $\xi$. By analytical and numerical calculations, we find that the ground state (GS) expectation value of the partial point group transformations behaves generically as $\langle \mathrm{GS}|g_D|\mathrm{GS}\rangle \sim \exp [i\theta +\gamma -\alpha \frac{\mathrm{Area}\left(\partial D\right)}{{\xi }^{d-1}}]$. Here, $\mathrm{Area}\left(\partial D\right)$ is the area of the boundary of the subregion $D$, and $\alpha$ is a dimensionless constant. The complex phase of the expectation value $\theta$ is quantized and serves as the topological invariant, and $\gamma$ is a scale-independent topological contribution to the amplitude. The examples we consider include the ${\mathbb{Z}}_8$ and ${\mathbb{Z}}_{16}$ invariants of topological superconductors protected by inversion symmetry in $(1+1)$ and $(3+1)$ dimensions, respectively, and the lens space topological invariants in $(2+1)$-dimensional fermionic topological phases. Connections to topological quantum field theories and cobordism classification of symmetry-protected topological phases are discussed.

Figure 1

(a) Fermionic matrix product representation of the partial reflection. (b) Path integral representation of the partial reflection. (c) One cross-cap on the torus.

Figure 2

Complex phase of the partial reflection $Z=\langle \mathrm{GS}|R_I|\mathrm{GS}\rangle$ for the disordered Kitaev Majorana chain. Each curve represents an ensemble average over 1000 samples. Solid lines are guides for the eye. Here, we set $\mathrm{\Delta }=t,N=200$, and $N_{\text{part}}=100$.

Figure 3

Complex phase of the partial reflection $Z=\langle \mathrm{GS}|R_I|\mathrm{GS}\rangle$ for one realization of the disorder potential. (a)–(f) represent different disorder strength from $W=1$ to 6 (same as the legend in Fig. 2). Solid lines are guides for the eye. Here, we set $\mathrm{\Delta }=t,N=200$, and $N_{\text{part}}=100$.

Figure 4

Phase diagram of the disordered Kitaev Majorana chain. (a) Color code is the Lyapunov exponent calculated using the transfer matrix approach (79), and (b) color code is the complex phase of the partial reflection $\langle \mathrm{GS}|R_I|\mathrm{GS}\rangle$. In panel (b), the red curve shows the phase boundary which is analytically determined by the transfer matrix as shown in (a). Here, we set $\mathrm{\Delta }=t,N=200$, and $N_{\text{part}}=100$.

Figure 5

(a) Partial rotation on the ground state $|\mathrm{\Psi }\rangle$. The figure shows the partial $C_4$ rotation. (b) A construction of lens space $L(n,1)$. The figure shows the boundary $\left(\cong S^2\right)$ of a 3-ball. The boundary of upper hemisphere is rotated by $2\pi /n$ angle, and glued into the boundary of lower hemisphere. The shadow regions are identified.

Figure 6

(a) Partial $C_4$ rotation on the square lattice and (b) partial $C_6$ rotation on the hexagonal lattice.

Figure 7

Complex phase $\angle Z=\mathrm{Im}ln\langle \mathrm{GS}|C_{n,D}|\mathrm{GS}\rangle$ of the partial rotation over (a) square lattice, and (b) hexagonal lattice. We set $t=\mathrm{\Delta }$. The total number of sites is $N={32}^2$ and the size of the transformed subsystem is $N_{\text{part}}={16}^2$ for (a) and $N_{\text{part}}=217$, where $R=8$ for (b).

Figure 8

Amplitude $\left|Z\right|=|\langle \mathrm{GS}|C_{n,D}|\mathrm{GS}\rangle |$ of (a) partial $C_4$ rotation on the square lattice (136), and (b) partial $C_6$ rotation on the hexagonal lattice (137) for various values of the chemical potential $\mu$. Here, we set $t=\mathrm{\Delta }$. The total number of sites is $N={32}^2$ and the dimension of the subsystem is given in terms of $L$ and $R$ (see Fig. 6) in each case. Solid lines are linear fits to data points.

Figure 9

(a) The scheme to extract topological contribution $\gamma$ in the partial fermion parity transformation (138). (b) Numerical results for the $(p_x-i{p}_y)$ superconductor [Eq. (136)] as a function of the chemical potential $\mu$. We set $t=\mathrm{\Delta }$. Here the total system size is $N={40}^2$ and the size of subsystem $A$ is $N_A={16}^2$.

Figure 10

Partial inversion on a ground state on 3d space torus $T^3$. The partial inversion transformation is performed only on inside of the 3-ball $D$ (the shadow region).

Figure 11

Complex phase of the partial inversion $\angle Z=\mathrm{Im}ln\langle \mathrm{GS}|I_D|\mathrm{GS}\rangle$ computed for 3D inversion symmetric topological superconductor (class D). (Top) I (II) corresponds to the phase with odd (even) number of gapless Majorana surface states. Here, we set $t=\mathrm{\Delta }$. The size of total system and subsystem are $N={12}^3$ and $N_{\text{part}}=6^3$, respectively.

Figure 12

The phase $\left(\angle Z\right)$ and modulus $\left(\left|Z\right|\right)$ of the expectation value of the partial inversion, $Z=\langle \mathrm{GS}|I_D|\mathrm{GS}\rangle$, computed for the 3D inversion symmetric topological insulator (class A) as a function of the mass parameter $m$ for various values of the $\mathrm{U}\left(1\right)$ phase transformation $b$ defined in Eq. (207). Strong (weak) TI refers to the phase with odd (even) number of Dirac surface states. Here, we set $t=r$. The size of total system and subsystem are $N={12}^3$ and $N_{\text{part}}=6^3$, respectively.

Figure 13

The phase $\left(\angle Z\right)$ and modulus $\left(\left|Z\right|\right)$ of the expectation value of the partial inversion, $Z=\langle \mathrm{GS}|I_D|\mathrm{GS}\rangle$, computed for the 3D inversion symmetric topological insulator as a function of the $\mathrm{U}\left(1\right)$ phase transformation $b$ defined in Eq. (207). Here, we set $t=r$ and $m=2$. The sizes of the whole system and the subsystem are $N={16}^3$ and $N_{\text{part}}=L^3$, respectively.

Figure 14

Wilson lines and the intersection forms.

Figure 15

The Wilson line and the intersection form on $\mathbb{R}{P}^2$.