Exact solution of the $D_3$ non-Abelian anyon chain

TL;DR

This paper provides an exact solution for a non-Abelian anyon chain based on the dihedral group D_3, connecting it to the XXZ Heisenberg model and analyzing its critical properties and phase diagram.

Contribution

It constructs commuting transfer matrices for the D_3 anyon chain and solves the spectral problem using Bethe ansatz, revealing connections to known spin chains and critical theories.

Findings

Spectrum related to XXZ spin-1/2 chain with boundary conditions

Identification of points realizing rational Z_2 orbifold theories

Finite size analysis of critical regime properties

Abstract

Commuting transfer matrices for linear chains of interacting non-Abelian anyons from the two-dimensional irreducible representation of the dihedral group $D_3$ (or, equivalently, the integer sector of the $su(2)_4$ spin-$1$ chain) are constructed using the spin-anyon correspondence to a $D_3$-symmetric formulation of the XXZ Heisenberg spin chain. The spectral problem is solved using discrete inversion identities satisfied by these transfer matrices and functional Bethe ansatz methods. The resulting spectrum can be related to that of the XXZ spin-$1/2$ Heisenberg chain with boundary conditions depending on the topological sector of the anyon chain. The properties of this model in the critical regime are studied by finite size analysis of the spectrum. In particular, points in the phase diagram where the anyon chain realizes some of the rational $\mathbb{Z}_2$ orbifold theories are identified.