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

TL;DR

The paper develops a general framework to detect interacting fermionic SPT phases protected by point-group and related symmetries via many-body invariants defined from partial symmetry operations on ground states. By linking ground-state overlaps under partial g_D to spacetime-path-integral constructs, the authors connect these invariants to cobordism classifications and TQFT data, with explicit calculations across (1+1)D, (2+1)D, and (3+1)D cases. They demonstrate that quantized complex phases in these overlaps reproduce known cobordism invariants (e.g., Z2, Z4, Z8, Z16) and lens-space or real-projective-space partition functions, often with characteristic topological amplitudes and area-law corrections. The work is supplemented by detailed lattice-model numerics validating the analytic predictions and by discussions of extensions to higher dimensions and to symmetry-enriched or topologically ordered phases. Overall, partial symmetry operations provide a versatile, nonperturbative tool to characterize SPT phases via many-body invariants consistent with cobordism and TQFT frameworks, with potential implications for experimental probes and numerical diagnostics.

Abstract

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. 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 $ξ$. By analytical and numerical calculations,we find that the ground state expectation value of the partial point group transformations behaves generically as $\langle GS | g_D | GS \rangle \sim \exp \Big[ i θ+ γ- α\frac{{\rm Area}(\partial D)}{ξ^{d-1}} \Big]$. Here, ${\rm Area}(\partial D)$ is the area of the boundary of the subregion $D$, and $α$ is a dimensionless constant. The complex phase of the expectation value $θ$ is quantized and serves as the topological invariant, and $γ$ 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.

Figures (15)

• 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: (Color online) Complex phase of the partial reflection $Z=\braket{GS | R_I | GS}$ 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 $\Delta=t$, $N=200$ and $N_\text{part}=100$.
• Figure 3: (Color online) Complex phase of the partial reflection $Z=\braket{GS | R_I | GS}$ for one realization of the disorder potential. Panels (a)-(f) represent different disorder strength from $W=1$ to $W=6$ (same as the legend in Fig. \ref{['fig:Kitaev_dis']}). Solid lines are guides for the eye. Here, we set $\Delta=t$, $N=200$ and $N_\text{part}=100$.
• Figure 4: (Color online) Phase diagram of the disordered Kitaev Majorana chain. (a) Color code is the Lyapunov exponent calculated using the transfer matrix approach (\ref{['eq:Tmat']}), and (b) color code is the complex phase of the partial reflection $\braket{GS | R_I | GS}$. 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 $\Delta=t$, $N=200$ and $N_\text{part}=100$.
• Figure 5: (a) Partial rotation on the ground state $\ket{\Psi}$. The figure shows the partial $C_4$ rotation. (b) A construction of lens space $L(n,1)$. The figure shows the boundary ($\cong S^2$) 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.