Generalized Kitaev models and extrinsic non-Abelian twist defects

We present a wide class of partially integrable lattice models with two-spin interactions which generalize the Kitaev honeycomb model. These models have a conserved quantity associated with each plaquette, conserved large loop operators on the torus, and topological degeneracy. We introduce a "slave-genon" approach which generalizes the Majorana fermion approach in the Kitaev model. The Hilbert space of our spin model can be embedded in an enlarged Hilbert space of non-Abelian twist defects, referred to as genons. In the enlarged Hilbert space, the spin model is exactly reformulated as a model of non-Abelian genons coupled to a discrete gauge field. We discuss in detail a particular Z_{3} generalization, and we show that in a certain limit the model is analytically tractable and produces a non-Abelian topological phase with chiral parafermion edge states.