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A study of the thermal properties of two-dimensional topological lattice models is presented. This work is relevant to assess the usefulness of these systems as a quantum memory. For our purposes, we use the topological mutual information $I_{mathrm{topo}}$ as a topological order parameter. For Abelian models, we show how $I_{mathrm{topo}}$ depends on the thermal topological charge probability distribution. More generally, we present a conjecture that $I_{mathrm{topo}}$ can (asymptotically) be written as a Kullback-Leitner distance between this probability distribution and that induced by the quantum dimensions of the model at hand. We also explain why $I_{mathrm{topo}}$ is more suitable for our purposes than the more familiar entanglement entropy $S_{mathrm{topo}}$. A scaling law, encoding the interplay of volume and temperature effects, as well as different limit procedures, are derived in detail. A non-Abelian model is next analysed and similar results are found. Finally, we also consider, in the case of a one-plaquette toric code, an environment model giving rise to a simulation of thermal effects in time.
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