Ground State Degeneracy in the Levin-Wen Model for Topological Phases

TL;DR

The paper demonstrates that ground-state degeneracy in the Levin-Wen model is a topological invariant determined solely by the surface topology, not lattice details. It develops a mutation-based framework using Pachner moves to show that ground-state subspaces are preserved under graph changes, enabling a general calculation of GSD from local data. In key examples, sphere GSD = 1 and SU_k(2) on a torus yields GSD = (k+1)^2, matching the predictions of doubled Chern-Simons theory and supporting the proposed equivalence with discrete TQFTs. The work thus provides a concrete lattice realization of doubled topological phases and lays groundwork for extensions to excitations and fluxon sectors.

Abstract

We study properties of topological phases by calculating the ground state degeneracy (GSD) of the 2d Levin-Wen (LW) model. Here it is explicitly shown that the GSD depends only on the spatial topology of the system. Then we show that the ground state on a sphere is always non-degenerate. Moreover, we study an example associated with a quantum group, and show that the GSD on a torus agrees with that of the doubled Chern-Simons theory, consistent with the conjectured equivalence between the LW model associated with a quantum group and the doubled Chern-Simons theory.

Figures (2)

• Figure 1: Given any two trivalent graphs $\Gamma^{(1)}$ and $\Gamma^{(2)}$ discretizing the same surface, we can always mutate $\Gamma^{(1)}$ to $\Gamma^{(2)}$ by a composition of elementary $f$ moves. In general $\Gamma^{(1)}$ and $\Gamma^{(2)}$ are not required to be regular lattices. These diagrams happen to be the same as Gu09, but in a slightly different context.
• Figure 2: All trivalent graphs can be reduced to their simplest structures by compositions of elementary $f$ moves. (a) on a sphere: 2 vertices, 3 edges, and 3 plaquettes. (b) on a torus: 2 vertices, 3 edges, and 1 plaquette.