Double band inversion in the topological phase transition of Ge1-x Snx alloys

We use first-principles simulation and virtual crystal approximation to reveal the unique double band inversion topological phase transition in Ge1-xSnx alloys. Wavefunction parity, spatial charge distribution surface state spectrum analyses suggest that is relayed by its first valence band. As system evolves from Ge {\alpha}-Sn, conduction moves down, inverts with second bands consecutively. The makes nontrivial, while one does not change invariant of system. Both inversions yield modes spanning individual inverted gaps, but only mode upper gap associates nontrivial nature tensile-strained {\alpha}-Sn.