Gábor B. Halász, Alioscia Hamma
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 R\'{e}nyi entropy of order 2. By exploiting an equivalence with the transverse field Ising model and establishing perturbation theories around the exactly solvable limits at zero and infinite fields, we determine the first three corrections to these quantities in the two limiting regimes. An exactly solvable variant of the problem is also studied to supplement the perturbation theories. We find strong evidence that the phase transition point between topological order and disorder is marked by a discontinuity in the topological R\'{e}nyi 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\'{e}nyi entropy is a proper topological invariant: its allowed values are discrete and can be used to distinguish between different phases of matter.
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http://arxiv.org/abs/1206.4223
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