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Band structure and topological phases of Pb$_{1-x-y}$Sn$_x$Mn$_y$Te by ab initio calculations

The change in the composition of Pb$_{1-x}$Sn$_x$Te IV-VI semiconductor or in its lattice parameter can drive a transition from the topologically trivial to the topological crystalline insulator (TCI), crossing a region where the alloy is in the Weyl semimetal phase. Incorporation of the magnetic Mn ions induces strong perturbations of the electronic structure, which act on both orbital and spin variables. Our first principles calculations show that the presence of Mn shifts the TCI and the Weyl region towards higher Sn contents in Pb$_{1-x}$Sn$_x$Te. When the Mn spin polarization is finite, the spin perturbation, like the orbital part, induces changes in band energies comparable to the band gap, which widens the Weyl area. The effect opens a possibility of driving transitions between various topological phases of the system by magnetic field or by the spontaneous Mn magnetization. We also propose a new method to calculate topological indices for systems with a finite spin polarization defined based on the concept of the Chern number. These valid topological characteristics enable an identification of the three distinct topological phases of the Pb$_{1-x-y}$Sn$_x$Mn$_y$Te alloy.