# On Defects Between Gapped Boundaries in Two-Dimensional Topological Phases of Matter

## TL;DR

This paper develops a theoretical framework to analyze the topological properties and projective braiding statistics of boundary defects in two-dimensional topological phases, with implications for quantum computation.

## Contribution

It introduces commuting Hamiltonians for boundary defects in 2+1D Dijkgraaf-Witten theories and establishes a bulk-edge correspondence using multi-fusion categories.

## Key findings

- Describes topological properties such as quantum dimensions of boundary defects.
- Elucidates projective braid statistics via bulk-edge correspondence.
- Analyzes boundary defects like Majorana, parafermion, genons, and in D(S_3).

## Abstract

Defects between gapped boundaries provide a possible physical realization of projective non-abelian braid statistics. A notable example is the projective Majorana/parafermion braid statistics of boundary defects in fractional quantum Hall/topological insulator and superconductor heterostructures. In this paper, we develop general theories to analyze the topological properties and projective braiding of boundary defects of topological phases of matter in two spatial dimensions. We present commuting Hamiltonians to realize defects between gapped boundaries in any $(2+1)D$ untwisted Dijkgraaf-Witten theory, and use these to describe their topological properties such as their quantum dimension. By modeling the algebraic structure of boundary defects through multi-fusion categories, we establish a bulk-edge correspondence between certain boundary defects and symmetry defects in the bulk. Even though it is not clear how to physically braid the defects, this correspondence elucidates the projective braid statistics for many classes of boundary defects, both amongst themselves and with bulk anyons. Specifically, three such classes of importance to condensed matter physics/topological quantum computation are studied in detail: (1) A boundary defect version of Majorana and parafermion zero modes, (2) a similar version of genons in bilayer theories, and (3) boundary defects in $\mathfrak{D}(S_3)$.