Lifshitz Transition in Topological Insulators with Second and Third-Neighbor Interactions

Topological Lifshitz transitions involve many types of topological structures in momentum and frequency-momentum spaces, where the energy band of a topological insulator is calculated taken into account second and third neighbors. A tight-binding model based on the Bernevig-Hughes Zhang (BHZ) approach for quantum wells is used to calculate the energies. The BHZ is characterized by the mass term M(k) = ∆ − Bk 2 . In the microscopic theory used here, the mass term is M(k) = ∆ − 2B(2 − cos(k x a) − cos(k y a)) that is modified when second and/or third neighbors are included in the model, as a consequence the range where the material is an insulator is changed.