Small correction to the quantization of Hall conductance due to current-current interactions and charge redistribution

The self-consistent Hall potential, current, charge, and magnetic-field distributions of an interacting electron gas with filled Landau levels in a thin strip are calculated. A Hartree approximation recently described by MacDonald, Rice, and Brinkman [Phys. Rev. B 28, 3648 (1983)] is extended to include the thickness of the strip and the effect of current-current interactions. It is found that the weighting of the Hall potential, current, and charge distributions toward the edges of the strip increases with decreasing thickness. This effect is due to the magnetic field generated by the current itself. The resulting self-consistent global charge and magnetic-field inhomogeneities lead to a nonvanishing correction to the quantization of the Hall conductance, which is nonlinear in the total current. For sample parameters corresponding to actual quantum Hall experiments the calculated correction is extremely small (one part in ${10}^{11}$).