Linearity of the edge states energy spectrum in the 2D topological insulator

Linearity of the topological insulator edge state spectrum plays a crucial role for various transport phenomena. Previous studies found that this linearity exists near the spectrum crossing point, but did not determine how perfect the linearity is. The purpose of the present study is to answer this question in various edge states models. We examine Volkov and Pankratov (VP) model for the Dirac Hamiltonian and the model BHZ for the Bernevig, Hughes and Zhang (BHZ) Hamiltonian with zero boundary conditions. It is found that both models yield ideally linear edge states. In the BHZ1 model the linearity is conserved up to the spectrum ending points corresponding to the tangency of the edge spectrum with the boundary of 2D states. In contrast, the model BHZ2 with mixed boundary conditions for BHZ Hamiltonian and the 2D tight-binding (TB) model yield weak nonlinearity.