A coupled wire description of surface $ADE$ topological orders

TL;DR

This paper develops a coupled wire framework to describe surface topological orders in 3D SPT/SET phases, focusing on Abelian orders classified by ADE Lie algebras, revealing symmetry and duality features.

Contribution

It introduces a coupled wire construction for surface topological orders in 3D SPT/SET phases, especially those with ADE classification, and explores their symmetry and duality properties.

Findings

Provides a coupled wire model for ADE-classified surface topological orders.

Elucidates emergent symmetry and duality in the coupled wire description.

Describes how gapped surface states relate to 1D gapless channels and their interactions.

Abstract

Symmetry-protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this paper, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the $ADE$ classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models.