SU($N$) Toric Code and Nonabelian Anyons

TL;DR

This paper develops an SU(N) toric code model on a 2D torus, revealing topologically distinct ground states and nonabelian anyons with braiding statistics described by Wigner rotation matrices.

Contribution

It introduces a new SU(N) toric code model with explicit construction of ground states and quasiparticles, highlighting nonabelian anyons and their braiding properties.

Findings

Identifies N^2 topologically distinct ground states.

Constructs all excited quasiparticle states with SU(N) charges.

Shows braiding statistics are encoded in Wigner rotation matrices.

Abstract

We construct SU($N$) toric code model describing the dynamics of SU($N$) electric and magnetic fluxes on a two dimensional torus. We show that the model has $N^2$ topologically distinct ground states $|\psi_0\rangle_{({\mathsf p},{\mathsf q})}$ which are loop states characterized by $Z_N \otimes Z_N$ centre charges $({\mathsf p},{\mathsf q} =0,1,2,\cdots, N-1)$. We explicitly construct them in terms of coherent superpositions of all possible spin network states on torus with Wigner coefficients as their amplitudes. All excited quasiparticle states with SU($N$) electric charges and magnetic fluxes are constructed. We show that the braiding statistics of these SU(N) electric, magnetic quasiparticles or nonabelian anyons is encoded in the Wigner rotation matrices.