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We calculate the electronic structure of the narrow gap semiconductors PbTe,SnTe and GeTe in the cubic phase using density functional theory (DFT) and the$G_0W_0$ method. Within DFT, we show that the band ordering obtained with aconventional semilocal exchange-correlation approximation is correct for SnTeand GeTe but wrong for PbTe. The correct band ordering at the high-symmetrypoint L is recovered adding $G_0W_0$ quasiparticle corrections. However,one-shot $G_0W_0$ produces artifacts in the band structure due to the wrongorbital character of the DFT single-particle states at the band edges close toL. We show that in order to correct these artifacts it is enough to considerthe off-diagonal elements of the $G_0W_0$ self-energy corresponding to thesestates. We also investigate the pressure dependence of the band gap for thesematerials and the possibility of a transition from a trivial to a non-trivialtopology of the band structure. For PbTe, we predict the band crossover andtopological transition to occur at around 4.8 GPa. For GeTe, we estimate thetopological transition to occur at 1.9 GPa in the constrained cubic phase, apressure lower than the one of the structural phase transition from rombohedralto cubic. SnTe is a crystalline topological insulator at ambient pressure, andthe transition into a trivial topology would take place under a volumeexpansion of approximately $10\%$.

$GW$ study of pressure-induced topological insulator transition in group IV-tellurides