Topological and nontopological degeneracies in generalized string-net models

TL;DR

This paper analyzes the energy-level degeneracies in generalized string-net models, revealing how noncommutative categories introduce non-topological degeneracies in excited states, expanding understanding of topological quantum phases.

Contribution

It provides a method to compute degeneracies in generalized string-net models, highlighting the role of internal multiplicities and noncommutative categories in excited state degeneracies.

Findings

Degeneracies depend on the Drinfeld center and internal multiplicities.

Noncommutative categories lead to extra nontopological degeneracies.

Results apply to any trivalent graph and orientable surface.

Abstract

Generalized string-net models have been recently proposed in order to enlarge the set of possible topological quantum phases emerging from the original string-net construction. In the present work, we do not consider vertex excitations and restrict to plaquette excitations, or fluxons, that satisfy important identities. We explain how to compute the energy-level degeneracies of the generalized string-net Hamiltonian associated to an arbitrary unitary fusion category. In contrast to the degeneracy of the ground state, which is purely topological, that of excited energy levels depends not only on the Drinfeld center of the category, but also on internal multiplicities obtained from the tube algebra defined from the category. For a noncommutative category, these internal multiplicities result in extra nontopological degeneracies. Our results are valid for any trivalent graph and any orientable surface. We illustrate our findings with nontrivial examples.