Self-duality and bound states of the toric code model in a transverse field

TL;DR

This paper studies how a transverse magnetic field affects the toric code model, revealing a self-dual mapping, phase transitions, and emergent bound states with unique dispersion properties.

Contribution

It introduces a mapping of the transverse field toric code to the quantum compass model and analyzes bound states and phase transitions.

Findings

Identification of a self-dual point and associated first order phase transition.

Emergence of bound states of quasiparticles with distinct statistics.

Good agreement with exact diagonalization except at the self-dual point.

Abstract

We investigate the effect of a transverse magnetic field on the toric code model. We show that this problem can be mapped onto the Xu-Moore model and thus onto the quantum compass model which are known to be self-dual. We analyze the low-energy spectrum by means of perturbative continuous unitary transformations and determine accurately the energy gaps of various symmetry sectors. Our results are in very good agreement with exact diagonalization data for all values of the parameters except at the self-dual point where level crossings are responsible for a first order phase transition between a topological phase and a polarized phase. Interestingly, bound states of two and four quasiparticles with fermionic and bosonic statistics emerge, and display dispersion relations of reduced dimensionality.