Anyon condensation in the string-net models

TL;DR

This paper develops a systematic method to describe anyon condensation of abelian bosons in string-net models, providing a way to transition between uncondensed and condensed phases and generalizing previous approaches to include chiral bosons.

Contribution

It introduces a comprehensive framework for condensing arbitrary abelian bosons in string-net models, extending previous methods to chiral and time-reversal invariant cases.

Findings

Constructed Hamiltonians for tuning through anyon condensation transitions.

Provided a systematic way to derive the condensed string-net data from uncondensed models.

Extended the framework to include chiral bosons and models without explicit time-reversal symmetry.

Abstract

We study condensation of abelian bosons in string-net models, by constructing a family of Hamiltonians that can be tuned through any such transition. We show that these Hamiltonians admit two exactly solvable, string-net limits: one deep in the uncondensed phase, described by an initial, uncondensed string net Hamiltonian, and one deep in the condensed phase, described by a final, condensed string net model. We give a systematic description of the condensed string net model in terms of the uncondensed string net and the data associated with the condensing abelian bosons. Specifically, if the uncondensed string net is described by a fusion category $\mathcal{C}$, we show how the string labels and fusion data of the fusion category $\mathcal{\tilde{C}}$ describing the condensed string net can be obtained from that of $\mathcal{C}$ and the data describing the string oeprators that create the condensing boson. This construction generalizes previous approaches to anyon condensation in string nets, by allowing the condensation of arbitrary abelian bosons, including chiral bosons in string nets constructed from (for example) Chern-Simons theories, which describe time-reversal invariant bilayer states. This gives a method for obtaining the full data for string nets without explicit time-reversal symmetry from such bilayer models. We illustrate our approach with several examples.