Long-range toric code: topological order at finite temperature in 2D

When topological order arises from short-ranged interactions in two dimensions, anyons are deconfined: the energy required to separate them arbitrarily far apart is finite. Consequently, thermally excited anyons proliferate and destroy topological order at any finite temperature. We introduce a toric code model with long-range interactions that preserves topological order at nonzero temperatures. Long-range interactions generate a confining potential between anyons, suppressing logical errors and stabilizing the topological phase against thermal fluctuations. For several forms of long-range couplings, we identify a confining topological phase below a critical temperature, where anyon excitations of the topologically ordered ground state remain confined. At the critical temperature, the system undergoes a transition into a trivial phase.