Integral quantum Hall effect for nonspecialists

An attempt is made to develop a description of the multielectron quantum state responsible for the integral quantum Hall effect. One goal is to provide intuitive support for the very powerful and general argument of Laughlin that the theoretical relationship is insensitive to complicating details in the interior of the sample. The model the author uses is somewhat more realistic than heretofore in that it is three dimensional, does not ignore the atomic structure of the bulk matter, and does not use an effective-mass approximation. In order to treat the problem quantum mechanically, the complete system, including circuitry external to the system of interest, is replaced by a model closed system consisting of a finite number of electrons. In this model, states with a finite Hall current and voltage are metastable against decay caused by interactions outside the model, such as those with bulk matter excitations. Such states describe the true situation well only in the conductivity plateaus; between plateaus, there would be current flow between the Hall voltage probes corresponding to decaying states. Experimental constraints replace this transverse current by a voltage drop along the direction of current flow. The interactions between the electrons are expressed in terms of a self-consistent potential which gives an independent-particle description as a starting point, and residual interactions which are treated by perturbation theory. The self-consistent potential is found to be important in understanding the properties of the quantum state of the system, such as the existence of the plateaus in conductivity and how the electrons in the (effective) two-dimensional region come to equilibrium with the different Fermi levels in the voltage probes. To all finite orders of perturbation theory, the residual interactions are found not to alter the quantized Hall conductivity.