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One of the key issues in the physics of topological insulators is whether thetopologically non-trivial 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\emph{ab-initio} methods supplemented by an effective dissipative theory builton the top of the \emph{ab-initio} electron and phonon band structures. In thecase of SnTe, the prototypical crystal topological insulator, we reveal thepresence of a characteristic temperature, much lower than the gap-closing one,that marks a loss of coherence of the topological state. The transition is notpresent in a purely electronic system but it appears once we invoke couplingwith a dissipative bosonic bath. Features in the dependence with temperature ofthe fidelity susceptibility can be related to changes in the band curvature,but signatures of a topological phase transition appear in the fidelity onlythough the non-adiabatic coupling with soft phonons. Our argument is valid forvalley topological insulators, but in principle can be generalized to thebroader class of topological insulators which host any symmetry-breaking boson.

Thermal topological phase transition in SnTe from \emph{ab-initio} calculations