Long range toric code: finite temperature topological order in two dimensions

In traditional 2D topological order, anyons quasiparticles are deconfined: the energy required to separate two anyons arbitrarily far apart is finite. As a result, thermally excited anyons proliferate and destroy topological order at any finite temperature. We introduce an extension of the toric code with long-range, power law interactions that maintains topological order at nonzero temperatures. The interactions generate a confining potential between anyons, 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 the ground state is topologically ordered but the excitations remain confined.