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

TL;DR

This paper introduces many-body topological invariants based on partial point group transformations to identify symmetry-protected topological phases in fermionic systems, providing analytical and numerical evidence for their effectiveness.

Contribution

It defines new topological invariants using partial point group transformations and demonstrates their quantized phases as indicators of topological phases in fermionic systems.

Findings

Quantized complex phase serves as a topological invariant.

Invariant detects $ ext{Z}_8$ and $ ext{Z}_{16}$ topological superconductors.

Numerical calculations confirm the theoretical predictions.

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 $\xi$. By analytical and numerical calculations,we find that the ground state expectation value of the partial point group transformations behaves generically as $\langle GS | g_D | GS \rangle \sim \exp \Big[ i \theta+ \gamma - \alpha \frac{{\rm Area}(\partial D)}{\xi^{d-1}} \Big]$. Here, ${\rm Area}(\partial D)$ 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.