| We discuss phase transitions between topologically ordered phases of matter and `ordinary' ones driven by either quantum mechanical perturbations or temperature, in the context of toric-code-like models. We show in particular that the topological entropy works indeed as an order parameter, and it is able to detect both types of transition. We also argue that robustness and stability at finite temperature are directly related to the confined / deconfined nature of the thermal excitations, which determines the relaxation time scales of a topologically ordered state. While the topological entropy is able to distinguish between the two behaviours, non-local winding operators vanish at finite temperature irrespective of the confined / deconfined nature of the excitations. Finally, we discuss how confinement in topologically ordered systems can be achieved either geometrically (e.g., in three or higher dimensions), or by means of effective mediated interactions (that obtain for example via coupling the toric code to phononic modes). |
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