Microscopic models for Kitaev's sixteenfold way of anyon theories

In two dimensions, the topological order described by Z₂ gauge theory coupled to free or weakly interacting fermions with a nonzero spectral Chern number is classified by 0. 28em{0ex}mod0. 28em{0ex}16 as predicted by Kitaev Ann. Phys. 321, 2 (2006). 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 matrices satisfying the Clifford algebra, enjoy a global SO () symmetry, and live on either square or honeycomb lattices depending on the parity of. 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 =2 and =3 models to materials with Kugel-Khomskii-type spin-orbital interactions is discussed.