Microscopic models for Kitaev's sixteenfold way of anyon theories

In two dimensions, the topological order described by ${\mathbb{Z}}_2$ gauge theory coupled to free or weakly interacting fermions with a nonzero spectral Chern number $\nu$ is classified by $\nu \mathrm{mod}16$ 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 $\mathrm{\Gamma }$ matrices satisfying the Clifford algebra, enjoy a global $\mathrm{SO}\left(\nu \right)$ 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.

Figure 1

Schematics of (a) square and (b) honeycomb lattices. The black (white) dots correspond to the $A$ ($B$) sublattice. The four (three) types of links on the square (honeycomb) lattice are denoted by different color codes. The square and hexagon illustrate the definition of the plaquette operators. The primitive vectors for the square (honeycomb) lattice are ${\mathbf{n}}_{1,2}=\left(\pm \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right) [{\mathbf{n}}_{1,2}=\left(\pm \frac{1}{2},\frac{\sqrt{3}}{2}\right)]$.

Figure 2

(a) Square lattice defined on a cylinder with operator $W_1$ for a noncontractible loop. (b) Ground state with an anyon flux $a$ threading the cylinder. (c) Spectrum of the itinerant Majorana fermions for the square lattice model on the cylinder with ground-state gauge configuration and periodic boundary condition ($W_1=1$), where the red dot indicates the Majorana zero modes. (d) The extracted conformal weight of the CFT primary field vs the cylinder circumference $L_1$ for $\nu =2$ (blue stars) and $\nu =3$ (red circles) models, where the horizontal dotted lines correspond to the expected values. The calculations in (c) and (d) are performed for cylinders of length $L_2=30$ [and the circumference $L_1=L_2$ in (c)] and employ $\kappa =0.2$.