J.M. Combes, F. Germinet
Edge and Impurity Effects on Quantization of Hall Currents
(309K, .pdf)

ABSTRACT.  We consider the edge Hall conductance and show it is invariant under perturbations located in a strip along the edge (decaying perturbations far from the edge are also allowed). This enables us to prove for the edge conductances a general sum rule relating currents due to the presence of two different media located respectively on the left and on the right half plane. As a particular interesting case we put forward a general quantization formula for the 
difference of edge Hall conductances in semi-infinite samples with and without 
a confining wall. It implies in particular that the edge Hall conductance takes its 
ideal quantized value under a gap condition for the bulk 
Hamiltonian, or under some localization properties for a random bulk Hamiltonian (provided one first regularizes the conductance; we shall discuss this regularization issue). Our quantization formula also shows that deviations from the ideal value occurs if a semi infinite distribution of 
impurity potentials is repulsive enough to produce current-carrying 
surface states on its boundary.