Hybridization and Field Driven Phase Transitions in Hexagonally Warped Topological Insulators

In this paper we discuss the role of material parameters and external field effects on a thin film topological insulator(TI) in the context of quantum phase transition(QPT). First, we consider an in-plane tilted magnetic field and determine the band structure of the surface states as a function of the tilt angle. We show that the presence of either a hybridization term or hexagonal warping or a combination of both leads to a semi-metal to insulator phase transition which is facilitated by their ${\cal PT}$ symmetry breaking character. We then note that while the introduction of an electric field does not allow for this QPT since it doesn't break ${\cal PT}$ symmetry, it can be used in conjunction with a tunneling element to reach a phase transition efficiently. The corresponding critical point is then non-trivially depend on the electric field, which is pointed out here. Then, we demonstrate that including a hexagonal warping term leads to an immediate ${\cal PT}$ symmetry violating QPT.