Aharonov-Bohm Effect for the Quantum Hall Conductivity on a Disordered Lattice

The Hall conductivity (${\sigma }_{\mathrm{xy}}$) on a lattice with disorder has been calculated in torus geometry to test the quantum Hall effect for finite magnetic fields $H$. It is shown that the quantum Hall effect persists independent of degree of disorder even away from the quantum ($H\to \infty$) limit, and that the Hall effect characteristic of periodic systems is not destroyed by disorder. The value of ${\sigma }_{\mathrm{xy}}$ for finite systems has a deviation from quantization, which depends on the Aharonov-Bohm flux $\Phi$ and disappears as the size is increased.