Liouvillian approach to the integer quantum Hall effect transition

Abstract

We present an approach to the localization-delocalization transition in the integer quantum Hall effect. The Hamiltonian projected onto the lowest Landau level can be written in terms of the projected density operators alone. This and the closed set of commutation relations between the projected densities leads to simple equations for the time evolution of the density operators. These equations can be used to map the problem of calculating the disorder-averaged and energetically unconstrained density-density correlation function to the problem of calculating the one-particle density of states of a dynamical system. At the self-consistent mean-field level, this approach yields normal diffusion and a finite longitudinal conductivity. While we have not been able to go beyond the saddle point approximation analytically, we show numerically that the critical localization exponent can be extracted from the energetically integrated correlation function, yielding v = 2.33 +/- 0.05, in excellent agreement with previous finite-size scaling studies.