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.

Figure 1

Magnetization $M$ versus loop tension. For small tension an almost constant susceptibility (see inset) leads to a linear increase of $M$. Above the critical loop tension (dashed line) the system approaches the fully polarized state.

Figure 2

Excitation gap for magnetic vortices (star symbols) versus loop tension. Right: Tunneling rate $\exp (-\beta \Delta )$ between topological sectors (circles) for $\beta =10$.

Figure 3

Energy splitting between topological sectors. The sector with $\{\mathrm{\text{even-odd}}/\mathrm{\text{odd-even}}\}$ winding number parities was taken as a reference (dashed line).

Figure 4

Finite-size scaling of the energy splitting between topological sectors around the phase transition for $\beta =10L$. At the critical loop tension (crosses) we find power-law scaling with exponent $z=1.42\pm 0.02$.

Figure 5

Confinement length of electric charge excitations. The confinement transition occurs at the same critical loop tension (dashed line) as the magnetic transition.

Figure 6

Gap and error probability versus dissipation strength of the dissipative Ising chain (4).