Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy

TL;DR

This paper develops criteria and mathematical tools to classify and analyze gapped domain walls and boundaries in 2+1D topologically ordered states, revealing their role in topological degeneracy and edge modes.

Contribution

It introduces the tunneling matrix as a new classification tool and provides criteria for the existence of gapped domain walls and gapless edge modes.

Findings

Tunneling matrices effectively classify gapped domain walls.

Criteria determine when topological orders must have gapless edges.

Derived a formula for topological ground state degeneracy with domain walls.

Abstract

Gapped domain walls, as topological line defects between 2+1D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which 2+1D topological orders must have gapless edge modes, namely which 1+1D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix $\mathcal W$, whose entries are the fusion-space dimensions $\mathcal W_{ia}$, to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.