Abstract: One of the key issues in the physics of topological insulators is whether the topologically nontrivial properties survive at finite temperatures and, if so, whether they disappear only at the temperature of topological gap closing. Here, we study this problem, using quantum fidelity as a measure, by means of ab initio methods supplemented by an effective dissipative theory built on the top of the ab initio electron and phonon band structures. In the case of SnTe, the prototypical crystal topological insulator, we reveal the presence of a characteristic temperature, much lower than the gap-closing one, that marks a loss of coherence of the topological state. The transition is not present in a purely electronic system but it appears once we invoke coupling with a dissipative bosonic bath. Features in the dependence with temperature of the fidelity susceptibility can be related to changes in the band curvature, but signatures of a topological phase transition appear in the fidelity only though the nonadiabatic coupling with soft phonons. Our argument is valid for valley topological insulators, but in principle can be generalized to the broader class of topological insulators which host any symmetry-breaking boson

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