Quantum Hall effect beyond the linear response approximation

Abstract

The problem of Bloch electrons in two dimensions subject to magnetic and intense electric fields is investigated, the quantum Hall conductance is calculated beyond the linear response approximation. Magnetic translations, electric evolution and energy translation operators are used to specify the solutions of the Schrödinger equation. For rational values of the magnetic flux quanta per unit cell and commensurate orientations of the electric field relative to the original lattice, an extended superlattice is defined and a complete set of mutually commuting space-time symmetry operators are obtained. Dynamics of the system is governed by a finite difference equation that exactly includes the effects of: an arbitrary periodic potential, an electric field orientated in a commensurable direction of the lattice, and coupling between Landau levels. A weak periodic potential broadens each Landau level in a series of minibands, separated by the corresponding minigaps; additionally the effect of the electric field in the energy spectrum is to superimpose equally spaced discrete levels, in this "magnetic Stark ladder" the energy separation is an integer multiple of $ h E / a B $, with $a$ the lattice parameter. A closed expression for the Hall conductance, valid to all orders in $\bs E$ is obtained, the leading order term reduces to the result of Thouless et al, in which $σ_H^{(0)}$ is quantized in units of $e^2/h$. The first order corrections exactly cancels for any miniband that is completely filled. Second order corrections for the miniband conductance are explicitly calculated as $σ_H^{(2)} \propto e^3/ U_0^2 B$, with $U_0$ the strength of the periodic potential. However the use of a sum rule shows that $σ_H^{(2)}$ cancels when a Landau band is fully occupied.