Anyons in an exactly solved model and beyond

TL;DR

This paper presents an exactly solvable honeycomb lattice spin model that exhibits both Abelian and non-Abelian anyons, with a detailed phase diagram and connections to topological invariants like the spectral Chern number.

Contribution

It introduces an exactly solvable model with tunable anyonic excitations and relates their properties to topological invariants, expanding understanding of topological quantum phases.

Findings

Identified phases with Abelian and non-Abelian anyons.

Connected anyonic properties to the spectral Chern number  and .

Demonstrated how magnetic fields induce non-Abelian anyons.

Abstract

A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static $\mathbb{Z}_{2}$ gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number $\nu$. The Abelian and non-Abelian phases of the original model correspond to $\nu=0$ and $\nu=\pm 1$, respectively. The anyonic properties of excitation depend on $\nu\bmod 16$, whereas $\nu$ itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.