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Topological phase transition and quantum spin Hall state in TlBiS$_2$

We have investigated the bulk and surface electronic structures and band topology of TlBiS$_2$ as a function of strain and electric field using \textit{ab-initio} calculations. In its pristine form, TlBiS$_2$ is a normal insulator, which does not support any non-trivial surface states. We show however that a compressive strain along the (111) direction induces a single band inversion with Z$_2$ = (1;000), resulting in a Dirac cone surface state with a large in-plane spin polarization. Our analysis shows that a critical point lies between the normal and topological phases where the dispersion of the 3D bulk Dirac cone at the $Γ$-point becomes nearly linear. The band gap in thin films of TlBiS$_2$ can be tuned through an out-of-the-plane electric field to realize a topological phase transition from a trivial insulator to a quantum spin Hall state. An effective $\mathbf{k \cdot p}$ model Hamiltonian is presented to simulate our first-principles results on TlBiS$_2$.