Topological robustness of quantization of the anomalous Hall conductance of a two-dimensional disordered Chern insulator

Abstract

The robustness of the anomalous Hall conductance, $\sigma_{yx}^{}= \sigma_{H}^{}$, quantization in the model of a~two-dimensional disordered gas of massive Dirac electrons subjected to an external orthogonal magnetic field is investigated in the framework of Kubo--St\v{r}eda formalism. Using the momentum representation for the averaged one-electron Green functions in a~magnetic field, an explicit analytical expression for the St\v{r}eda term of $\sigma_{H}^{}$ is obtained. It is shown that this term is proportional to the topological Chern number, ${\rm Ch}=\pm 1/2$, if the Fermi level is in the energy gap. In this case, the total $\sigma_{H}^{}$ takes the half-integer quantized value, $\sigma_{H}^{}=\pm e_{}^{2}/4\pi\hbar$,that does not depend on either the magnitude of the disorder or the strength of the external magnetic field. As an example, we calculated the densities of states and intrinsic anomalous Hall conductance, $\sigma_{H}^{\rm int}$, of a~two-dimensional disordered gas of massive Dirac electrons subjected to an external magnetic field in the self-consistent Born approximation. A numerical analysis of the field and energy dependencies of these expressions is carried out for various values of the parameters of the model under consideration. In particular, the results of this analysis show that the St\v{r}eda term of $\sigma_{H}^{}$ is susceptible to disorder in a~sufficiently wide vicinity of the point of transition to the quantization regime.