Breakdown of a Topological Phase: Quantum Phase Transition in a Loop Gas Model with Tension

We study the stability of topological order against local perturbations by considering the effect of a magnetic field on a spin model—the toric code—which is in a topological phase. The model can be mapped onto a quantum loop gas where the perturbation introduces a bare loop tension. When the loop tension is small, the topological order survives. When it is large, it drives a continuous quantum phase transition into a magnetic state. The transition can be understood as the condensation of “magnetic” vortices, leading to confinement of the elementary “charge” excitations. We also show how the topological order breaks down when the system is coupled to an Ohmic heat bath and relate our results to error rates for topological quantum computations.