Topological Flux Phases of Levin-Wen String-Net Models

Levin-Wen string-net models provide exactly-solvable lattice models for gapped topological phases. We examine flux phases of these models, in which the lattice plaquettes contain a nontrivial flux instead of containing zero flux. In particular, we study $Z_N$ and Ising flux phases. We find that the Ising $\sigma$ flux phase is gapless, but nonetheless contains quasiparticles with topologically protected non-Abelian braiding statistics, thus providing an exactly-solvable model of a quasi-topological phase.