Quantized Hall conductance: Effect of random potentials

Abstract

The effect of random potentials on the Hall current is expressed in terms of a sum of level shift functions η(ɛ). These functions obey a generalized Levinson theorem:\(\eta (\bar \varepsilon ) - \eta (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varepsilon } ) = \pi N_b \) = number of localized states where\(\bar \varepsilon \) and\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varepsilon } \) are the upper and lower mobility edge respectively of a band of extended states. As a consequence of that the total Hall current carried by a band of extended states originating from a Landau level remains unaffected by a random potential independent of the number of bound states split off from the continuum.