  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
non-local quantities: the Wilson loop and the topological Renyi 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 Renyi 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 Renyi entropy is a proper
topological invariant: its allowed values are discrete and can be used to
distinguish between different phases of matter.
