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

TL;DR

The paper addresses how to determine whether a (2+1)D symmetry-protected topological (SPT) edge theory can be gapped without breaking the protecting symmetry by analyzing modular invariance of the symmetry-projected (orbifold) edge conformal field theories. It develops a framework that treats SPT edges as orbifolds, requiring modular invariance of the twisted partition functions to signal a trivial gappable phase, while modular noninvariance signals a nontrivial bulk. For Abelian bosonic SPTs with $ obrule Z_K imes obrule Z_K$ symmetry, modular invariance occurs only when the number of edge flavors satisfies $N_f mod K = 0$, enabling a symmetry-preserving gapping potential; for fermionic SPTs, the condition is $N_f mod (2K)$ for even $K$ and $N_f mod K$ for odd $K$, with explicit gapping constructions provided. The approach offers a nonperturbative, interaction-based diagnostic that connects Laughlin-like flux threading to modular transformations and extends to SETs and non-Abelian contexts, enabling broad classification and potential experimental relevance.

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.