A remark on gapped domain walls between topological phases

TL;DR

This paper provides a rigorous mathematical framework for gapped domain walls and boundaries in topological phases, addressing recent physics questions and clarifying conjectures with precise definitions and counterexamples.

Contribution

It introduces a formal mathematical definition of gapped domain walls and boundaries, and resolves open questions by analyzing and refuting a conjecture from condensed matter physics.

Findings

Identified the tunneling matrix and coupling matrix in topological phases.

Provided a mathematical counterexample to a recent conjecture.

Clarified the relationship between physical and mathematical descriptions of topological boundaries.

Abstract

We give a mathematical definition of a gapped domain wall between topological phases and a gapped boundary of a topological phase. We then provide answers to some recent questions studied by Lan, Wang and Wen in condensed matter physics based on works of Davydov, M\"uger, Nikshych and Ostrik. In particular, we identify their tunneling matrix and a coupling matrix of Rehren, and show that their conjecture does not hold.