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

Understand

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 $h_x^{(c)}=h_z^{(c)}=0.418(2)$.