  By studying the quantum Hall effect of stationary states with high values of
injected current using a von Neumann lattice representation, we found that
broadening of extended state bands due to a Hall electric field occurs and
causes the breakdown of the quantum Hall effect. The Hall conductance agrees
with a topological invariant that is quantized exactly below a critical field
and is not quantized above a critical field. The critical field is proportional
to $B^{3/2}$ and is enhanced substantially if the extended states occupy a
small fraction of the system.
