Braiding statistics and classification of two-dimensional charge-$2m$ superconductors

TL;DR

This paper classifies the braiding statistics and topological phases of two-dimensional charge-$2m$ superconductors coupled to a $ ext{Z}_{2m}$ gauge field, revealing a dependence on the parity of $m$ and identifying the absence of certain SPT phases.

Contribution

It provides a comprehensive classification of braiding statistics and topological phases for charge-$2m$ superconductors with $ ext{Z}_{2m}$ symmetry, including new insights into the absence of certain fermionic SPT phases.

Findings

16m braiding types for odd m

4m braiding types for even m

No nontrivial fermionic SPT phase with Z4^f symmetry

Abstract

We study braiding statistics between quasiparticles and vortices in two-dimensional charge-$2m$ (in units of $e$) superconductors that are coupled to a $\mathbb Z_{2m}$ dynamical gauge field, where $m$ is any positive integer. We show that there exist $16m$ types of braiding statistics when $m$ is odd, but only $4m$ types when $m$ is even. Based on the braiding statistics, we obtain a classification of topological phases of charge-$2m$ superconductors---or formally speaking, a classification of symmetry-protected topological phases, as well as invertible topological phases, of two-dimensional gapped fermions with $\mathbb Z_{2m}^f$ symmetry. Interestingly, we find that there is no nontrivial fermionic symmetry-protected topological phase with $\mathbb Z_4^f$ symmetry.