Phase diagram of the toric code model in a parallel magnetic field

TL;DR

This paper maps the phase diagram of the toric code model under a parallel magnetic field, identifying three phases and characterizing phase transitions using Monte Carlo simulations.

Contribution

It introduces an implicit local order parameter and provides numerical analysis of phase transitions in the toric code model under magnetic fields.

Findings

Identifies three distinct phases: topological, charge-condensed, vortex-condensed.

Determines the second-order transition line via finite-size scaling.

Locates the endpoint of the first-order transition line at specific magnetic field values.

Abstract

Ground-state phase diagram of the toric code model in a parallel magnetic field has three distinct phases: topological, charge-condensed, and vortex-condensed states. To study it we consider an implicit local order parameter characterizing the transition between the topological and charge-condensed phases, and sample it using continuous-time Monte Carlo simulations. The corresponding second-order transition line is obtained by finite-size scaling analysis of this order parameter. Symmetry breaking between charges and vortices along the first-order transition line is also observed, and our numerical result shows that the end point of the first-order transition line is located at $h_x^{(c)}=h_z^{(c)}=0.418(2)$.