Thermal topological phase transition in SnTe from ab initio calculations

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.

Figure 1

Detail of the band structure around the $L$ point showing the evolution of the band structure of SnTe with temperature, as calculated with DFT (the band structure along a full path across the Brillouin zone can be found in the Supplemental Material [14]). The temperature $T$ varies from 100 to 1900 K in 200-K steps. Inset: Group velocities of electrons in the conduction band as a function of temperature.

Figure 2

Fidelity as a function of temperature computed from the single-particle DFT density matrix. Data for temperatures above the melting point, and therefore not experimentally accessible, are plotted as a dashed line.

Figure 3

An overall $T$ dependence of quantum fidelity derivative (with a minus sign) obtained by truncating the interactions at $(n=10)\text{th}$ order and integrating over the entire BZ to find $d{\rho }_{\text{inc}}^{\left(n\right)}\left(T\right)/dT\sim {\chi }_F\left(T\right)$. We see a peak indicating the phase transition.

Figure 4

Momentum-resolved fidelity susceptibility ${\chi }_{\mathcal{F}}\left(T\right)$ shown in a region of the BZ around the $L$ point. In the main panel we show isosurfaces corresponding to ${\chi }_{\mathcal{F}}\left(T\right)=0.02$ for $T=300$ K (blue), 500 K (pink), and 900 K (green). In the first two cases there are two minima and for 500 K the isosurface is the largest. The inset shows three isosurfaces ${\chi }_{\mathcal{F}}\left(T\right)=0.3, {\chi }_{\mathcal{F}}\left(T\right)=0.03, {\chi }_{\mathcal{F}}\left(T\right)=0.01$ for $T=500$ K, showing that the largest values of drag are indeed concentrated close to the minima.