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Superconductivity in topological insulators is expected to show veryunconventional features such as $p+ip$ order parameter, Majorana fermions etc.However, the intrinsic superconductivity has been observed in a very limitednumber of materials in which the pairing symmetry is still a matter of debate.Here, we study the topological crystalline insulator(Pb$_{1-x}$Sn$_x$)$_{1-y}$In$_y$Te, where a peculiar insulator tosuperconductor transition was previously reported near the gap inversiontransition, when the system is nearly a 3-dimensional Dirac semimetal. Both theexistence of superconductivity near the 3-dimensional Dirac semimetal and theoccurrence of insulator to superconductor transition in an isotropic materialis highly unusual. We suggest that the observed phenomena are related to anintrinsic instability of a 3-dimensional Dirac semimetal state in(Pb$_{1-x}$Sn$_x$)$_{1-y}$In$_y$Te and "flattening" of the bulk valence andconduction bands as they acquire a Mexican hat-like dispersion on the invertedside of the phase diagram. This favors the pairing instability if the chemicalpotential is pinned to these flat regions.

Possible Topological Superconductivity in a Topological Crystalline Insulator (Pb$_{1-x}$Sn$_x$)$_{1-y}$In$_y$Te