Born approximation study of the strong disorder in magnetized surface states of a topological insulator

In this study, we investigate the effect of random point disorder on the surface states of a topological insulator with out-of-plane magnetization. We consider the disorder within a high-order Born approximation. The Born series converges to the one branch of the self-consistent Born approximation (SCBA) solution at low disorder. As the disorder strength increases, the Born series converges to another SCBA solution with the finite density of states within the magnetization-induced gap. Further increase of the disorder strength leads to a divergence of the Born series, showing the limits of the applicability of the Born approximation. We find that the convergence properties of this Born series are closely related to the properties of the logistic map, which is known as a prototypical model of chaos. We also calculate the longitudinal and Hall conductivities within the Kubo formulas at zero temperature with the vertex corrections for the velocity operator. Vertex corrections are important for describing transport properties in the strong disorder regime. In the case of a strong disorder, the longitudinal conductivity is weakly dependent on the disorder strength, while the Hall conductivity decreases with increasing disorder.