Ground State Degeneracy of Topological Phases on Open Surfaces

TL;DR

Develops a framework to compute the ground-state degeneracy $GSD$ of non-Abelian topological orders on open surfaces via anyon condensation and gapped boundaries, where each boundary is labeled by a generalized Lagrangian subset $L$ and the bulk $GSD$ equals the number of confined anyons in $\mathcal{T}_L$. It validates the approach on examples including the $\\mathbb{Z}_2$ toric code and the $\\mathbb{Z}_2^3$ twisted quantum double, illustrating both confinement-based and charge-transport (condensate-pumping) viewpoints and the folding relation to doubled theories. A key refinement is that condensed anyons may split into multiple vacua, requiring multiplicity data in $L$ to correctly count $GSD$, as demonstrated in the $L_D$ case. The results connect to generalized Laughlin–Wu–Tao pumping for non-Abelian phases and suggest practical routes to engineer desired conserved bulk anyons for topological 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.

Figures (4)

• Figure 1: Ground state basis specified by a conserved anyon line $m$ winding the cylinder where the boundaries are characterized by the $L_e$ condensate. A unit of $e$ is transferred across the boundaries via the LTW charge pumping mechanism if the unit $m$ line can be changed to two units adiabatically.
• Figure 2: (a) $\text{Fibo}\times\overline{\text{Fibo}}$ on a cylinder with both boundaries characterized by the Lagrangian subset $\{1,\tau\bar{\tau}\}$. (b) Single Fibonacci phase on a torus. The two systems are equivalent.
• Figure 3: (a) An open surface with $M>3$ boundaries. (b) A genus-$(M-1)$ torus.
• Figure 4: $\mathbb{Z}_2^3$ twisted quantum double on a cylinder with the left boundary characterized by electric condensation $L_{\textrm{E}}$ and the right one by dyonic condensation $L_{\textrm{D}}$. $GSD=|T_{\textrm{E}}\cap T_{\textrm{D}}|$.