Realizing non-Abelian statistics in time-reversal-invariant systems

We construct a series of $(2+1)$-dimensional models whose quasiparticles obey non-Abelian statistics. The adiabatic transport of quasiparticles is described by using a correspondence between the braid matrix of the particles and the scattering matrix of $(1+1)$-dimensional field theories. We discuss in depth lattice and continuum models whose braiding is that of SO(3) Chern-Simons gauge theory, including the simplest type of non-Abelian statistics, involving just one type of quasiparticle. The ground-state wave function of an SO(3) model is related to a loop description of the classical two-dimensional Potts model. We discuss the transition from a topological phase to a conventionally ordered phase, showing in some cases there is a quantum critical point.

Figure 1

(Color online) Braid, reverse braid, and Temperley-Lieb generator.

Figure 2

(Color online) A typical braiding involving six particles.

Figure 3

(Color online) Consistency relation for braiding.

Figure 4

(Color online) The Temperley-Lieb algebra.

Figure 5

(Color online) The braid in terms of the Temperley-Lieb generator.

Figure 6

(Color online) Projecting onto the spin-1 representation.

Figure 7

(Color online) The generators of the SO(3) BMW algebra.

Figure 8

(Color online) A typical configuration in the spin-1 loop model.

Figure 9

(Color online) Two trivalent vertices in the SO(3) BMW algebra.

Figure 10

(Color online) Removing one loop in the spin-1 loop model.

Figure 11

(Color online) A trivalent vertex in the three-state Potts model.

Figure 12

(Color online) The recursion relation for the chromatic polynomial.

Figure 13

(Color online) Plaquettes with two external occupied links.

Figure 14

(Color online) Plaquettes with three external occupied links.

Figure 15

(Color online) Plaquettes with four external occupied links.

Figure 16

(Color online) Removing or adding an isolated loop.

Figure 17

(Color online) Graphical representations of two configurations.

Figure 18

(Color online) Removing one loop of a four-hexagon cluster.