Ground State Degeneracy of Topological Phases on Open Surfaces

TL;DR

This paper explores how the ground state degeneracy of non-Abelian topological phases on surfaces with boundaries relates to anyon condensates, providing a framework for understanding boundary conditions and potential quantum computing applications.

Contribution

It establishes a one-to-one correspondence between gapped boundary conditions and anyon condensates, generalizing the charge-pumping argument to non-Abelian phases.

Findings

Gapped boundary conditions correspond to specific anyon condensates.

The ground state degeneracy equals the number of confined sectors due to condensation.

Generalized charge-pumping can manipulate anyons for quantum computation.

Abstract

We relate the ground state degeneracy (GSD) of a non-Abelian topological phase on a surface with boundaries to the anyon condensates that break the topological phase to a trivial phase. Specifically, we propose that gapped boundary conditions of the surface are in one-to-one correspondence to the sets of condensates, each being able to completely break the phase, and we substantiate this by examples. The GSD resulting from a particular boundary condition coincides with the number of confined topological sectors due to the corresponding condensation. These lead to a generalization of the Laughlin-Wu-Tao (LWT) charge-pumping argument for Abelian fractional quantum Hall states (FQHS) to encompass non-Abelian topological phases, in the sense that an anyon loop of a confined anyon winding a non-trivial cycle can pump a condensate from one boundary to another. Such generalized pumping may find applications in quantum control of anyons, eventually realizing topological quantum computation.