Microscopic models for Kitaev's sixteenfold way of anyon theories

In two dimensions, the topological order described by Z(2) gauge theory coupled to free or weakly interacting fermions with a nonzero spectral Chem number nu is classified by nu mod 16 as predicted by Kitaev [Ann. Phys. 321. 2 (20064 Here, we provide a systematic and complete construction of microscopic models realizing this so-called sixteenfold way of anyon theories. These models are defined by Gamma matrices satisfying the Clifford algebra, enjoy a global SO(nu) symmetry, and live on either square or honeycomb lattices depending on the parity of nu. We show that all these models are exactly solvable by using a Majorana representation and characterize the topological order by calculating the topological spin of an anyonic quasiparticle and the ground-state degeneracy. The possible relevance of the nu = 2 and nu = 3 models to materials with Kugel-Khomskii-type spin-orbital interactions is discussed.