Invariants for (2+1)D bosonic crystalline topological insulators for all 17 wallpaper groups

TL;DR

This paper develops a set of many-body invariants to detect and classify all (2+1)D bosonic crystalline topological insulators across all 17 wallpaper groups, using symmetry operations like partial rotations and reflections.

Contribution

It introduces a comprehensive framework of invariants based on symmetry operations to fully characterize bosonic crystalline topological phases in (2+1)D.

Findings

Proposes invariants for all wallpaper groups and internal symmetries.

Verifies invariants through exact ground state calculations.

Derives a topological effective action for orientation-reversing symmetries.

Abstract

We study bosonic symmetry-protected topological (SPT) phases in (2+1) dimensions with symmetry $G = G_{\text{space}}\times K$, where $G_{\text{space}}$ is a general wallpaper group and $K=\text{U}(1),\mathbb{Z}_N, \text{SO}(3)$ is an internal symmetry. In each case we propose a set of many-body invariants that can detect all the different phases predicted from real space constructions and group cohomology classifications. They are obtained by applying partial rotations and reflections to a given ground state, combined with suitable operations in $K$. The reflection symmetry invariants that we introduce include `double partial reflections', `weak partial reflections' and their `relative' or `twisted' versions which also depend on $K$. We verify our proposal through exact calculations on ground states constructed using real space constructions. We demonstrate our method in detail for the groups p4m and p4g, and in the case of p4m also derive a topological effective action involving gauge fields for orientation-reversing symmetries. Our results provide a concrete method to fully characterize (2+1)D crystalline topological invariants in bosonic SPT ground states.