Folding approach to topological orders enriched by mirror symmetry

TL;DR

This paper introduces a folding method to analyze two-dimensional mirror symmetry-enriched topological phases, transforming complex nonlocal symmetries into manageable boundary properties, thus facilitating the study of anomalies and classifications.

Contribution

The paper develops a folding approach that maps mirror SETs to boundary properties, enabling the use of anyon condensation theory to analyze anomalies and classify mirror-enriched topological phases.

Findings

Derived physical constraints on mirror permutation and fractionalization

Reproduced known classification and anomaly results

Proposed conjecture on the completeness of the constraints

Abstract

We develop a folding approach to study two-dimensional symmetry-enriched topological (SET) phases with the mirror reflection symmetry. Our folding approach significantly transforms the mirror SETs, such that their properties can be conveniently studied through previously known tools: (i) it maps the nonlocal mirror symmetry to an onsite $\mathbb{Z}_2$ layer-exchange symmetry after folding the SET along the mirror axis, so that we can gauge the symmetry; (ii) it maps all mirror SET information into the boundary properties of the folded system, so that they can be studied by the anyon condensation theory---a general theory for studying gapped boundaries of topological orders; and (iii) it makes the mirror anomalies explicitly exposed in the boundary properties, i.e., strictly 2D SETs and those that can only live on the surface of a 3D system can be easily distinguished through the folding approach. With the folding approach, we derive a set of physical constraints on data that describes mirror SET, namely mirror permutation and mirror symmetry fractionalization on the anyon excitations in the topological order. We conjecture that these constraints may be complete, in the sense that all solutions are realizable in physical systems. Several examples are discussed to justify this. Previously known general results on the classification and anomalies are also reproduced through our approach.