Symmetry-protected topological phases and orbifolds: Generalized Laughlin's argument

TL;DR

This paper investigates how modular invariance of edge theories in 2D symmetry-protected topological phases can determine whether edges can be gapped without breaking symmetry, using Chern-Simons and topological superconductor models.

Contribution

It introduces a method to diagnose edge gappability in SPT phases via modular invariance analysis, applicable to both bosonic and fermionic systems.

Findings

Modular invariance indicates the possibility of symmetric edge gapping.

Explicit construction of symmetric potentials that gap edges when invariance is achieved.

Demonstration with Chern-Simons and topological superconductor models.

Abstract

We consider non-chiral symmetry-protected topological phases of matter in two spatial dimensions protected by a discrete symmetry such as $\mathbb{Z}_K$ or $\mathbb Z_K \times \mathbb Z_K $ symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as an example, we demonstrate explicitly that when the modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge.