Low-energy effective theory of the toric code model in a parallel field

TL;DR

This paper analytically maps the phase diagram of the toric code model in a parallel magnetic field, revealing phase transitions, quasiparticle behavior, and critical properties using perturbative and large-spin methods.

Contribution

It provides an analytical determination of the phase diagram and critical behavior of the toric code in a parallel field, complementing previous numerical studies.

Findings

Identification of three distinct phases in the phase diagram.

Discovery of two second-order transition lines merging into a first-order line at a multicritical point.

Calculation of critical fields, exponents, and quasiparticle dispersions.

Abstract

We determine analytically the phase diagram of the toric code model in a parallel magnetic field which displays three distinct regions. Our study relies on two high-order perturbative expansions in the strong- and weak-field limit, as well as a large-spin analysis. Calculations in the topological phase establish a quasiparticle picture for the anyonic excitations. We obtain two second-order transition lines that merge with a first-order line giving rise to a multicritical point as recently suggested by numerical simulations. We compute the values of the corresponding critical fields and exponents that drive the closure of the gap. We also give the one-particle dispersions of the anyonic quasiparticles inside the topological phase.