Pressure-Induced Topological Phase Transitions in CdGeSb$_2$ and CdSnSb$_2$

Topological quantum phase transitions (TQPTs) in a material induced by external perturbations are often characterized by band touching points in the Brillouin zone. The low-energy excitations near the degenerate band touching points host different types of fermions while preserving the topological protection of surface states. An interplay of different tunable topological phases offers an insight into the evolution of the topological character. In this paper, we study the occurrence of TQPTs as a function of hydrostatic pressure in CdGeSb$_2$ and CdSnSb$_2$ chalcopyrites, using the first-principles calculations. At ambient pressure, both materials are topological insulators having a finite band gap with inverted order of Sb-$s$ and Sb-$p_x$,$p_y$ orbitals of valence bands at the $\Gamma$ point. On the application of hydrostatic pressure the band gap reduces, and at the critical point of the phase transition, these materials turn into Dirac semimetals. On further increasing the pressure beyond the critical point, the band inversion is reverted making them trivial insulators. The pressure-induced change in band topology from non-trivial to trivial phase is also captured by L\"{u}ttinger model Hamiltonian calculations. Our model demonstrates the critical role played by a pressure-induced anisotropy in frontier bands in driving the phase transitions. These theoretical findings of peculiar coexistence of multiple topological phases in the same material provide a realistic and promising platform for the experimental realization of the TQPT.