Magnetic field control of Weyl node population in Pb$_{1-x}$Sn$_x$Te

For materials possessing topological phase transition, the Weyl semimetal phase can be induced by breaking either the time-reversal or inversion symmetry. The topological crystalline insulator, Pb$_{1-x}$Sn$_x$Te exhibits topological phase transition upon the band inversion strength which can be tailored by the substitutional mixing ratio, strain, thermal expansion, ferroelectric displacement, and/or material thickness via quantum confinement effect. The SnTe building block of the compound is also known to exhibit a ferroelectric transition at low temperatures which leads to inversion symmetry breakdown. Therefore one can expect that Pb$_{1-x}$Sn$_x$Te exhibits diverse topological phases including a Weyl semimetal phase. In this study, using $\it{ab}$-$\it{initio}$-tight-binding calculations we have explored the parameter space associated with both band inversion and ferroelectric displacement. The calculated topological phase diagram shows the emergence of a Weyl semimetal phase. We will also present results of the evolution of Weyl nodes with magnetic field.