Exactly soluble lattice models for non-Abelian states of matter in two dimensions

Following an earlier construction of exactly soluble lattice models for Abelian fractional topological insulators in two and three dimensions, we construct here an exactly soluble lattice model for a non-Abelian $\nu =1$ quantum Hall state and a non-Abelian topological insulator in two dimensions. We show that both models are topologically ordered, exhibiting fractionalized charge, ground-state degeneracy on the torus, and protected edge modes. The models feature non-Abelian vortices which carry fractional electric charge in the quantum Hall case and spin in the topological insulator case. We analyze the statistical properties of the excitations in detail and discuss the possibility of extending this construction to three-dimensional non-Abelian topological insulators.

Figure 1

The Hilbert spaces are on the links, which we have denoted with orange dots; we have suppressed link orientations in the picture. The operators $B$ act on the plaquettes, while the operators $A$ act the on stars. A combination of a plaquette $P$ and one of its vertices $s$ is a site $x$.

Figure 2

(a) The ribbon operator between the sites $x_0$ and $x_1$; the arrow denotes the direction of the ribbon. The affected links (Hilbert spaces) have been colored red. (b) Elementary ribbon operators. Every longer ribbon can be composed of elementary ones. The mutual orientations in the left row are defined as positive.

Figure 3

Ribbon operators creating excitations at sites $x_1,...,x_4$.

Figure 4

Ribbon operators creating excitations at sites $x_1,...,x_4$ anchored at an auxiliary site $x_0$.

Figure 5

The exchange of particles at sites $i$ and $i+1$ effected by the $R$ operator. The respective ribbon operators have to be commuted.