Hall conductivity of two-dimensional electrons in a periodic potential

Abstract

Hall conductivity of two-dimensional electrons in a weak periodic potential $V\left(r\right)\mathrm{}$ and in a strong magnetic field $H, \mathrm{}\left|V\right(r\left)\right|\ll \hslash {\omega }_c=\frac{\hslash \mathrm{eH}}{\mathrm{mc}}$, is calculated by the Kubo formula. In the presence of $V\left(r\right)\mathrm{}$, each Landau level splits into several energy bands. The Hall conductivity takes a quantized value, a multiple of $\frac{e^2}{h}$, whenever the Fermi level is in an energy gap in accordance with the general argument of the quantized Hall effect. These results are used to discuss the Hall conductivity of the charge-density-wave state in the presence of a weak random potential.