Probing topological order with Rényi entropy

We present an analytical study of the quantum phase transition between the topologically ordered toric-code-model ground state and the disordered spin-polarized state. The phase transition is induced by applying an external magnetic field, and the variation in topological order is detected via two nonlocal quantities: the Wilson loop and the topological R\'enyi entropy of order 2. By exploiting an equivalence with the transverse-field Ising model and considering two different variants of the problem, we investigate the field dependence of these quantities by means of an exact treatment in the exactly solvable variant and complementary perturbation theories around the limits of zero and infinite fields in both variants. We find strong evidence that the phase transition point between topological order and disorder is marked by a discontinuity in the topological R\'enyi entropy and that the two phases around the phase transition point are characterized by its different constant values. Our results therefore indicate that the topological R\'enyi entropy is a proper topological invariant: its allowed values are discrete and can be used to distinguish between different phases of matter.