We construct a mapping between the two-dimensional toric code model in external magnetic fields, $h_z$ and $h_x$, and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional ${\mathbb{Z}}_2$ gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large (up to ${60}^3$) lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the ${\mathbb{Z}}_2$ gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.

• Received 14 June 2010

DOI:https://doi.org/10.1103/PhysRevB.82.085114