Anderson localization in Liouville space: The effective dephasing approximation

The effective dephasing approximation (EDA) provides a self-consistent procedure for calculating the transport properties of a quantum particle in a disordered medium. It is based on mapping the averaged Liouville-space propagator into the propagator of a particle moving in an ordered lattice with an effective frequency-dependent dephasing rate. The effective dephasing rate is determined self-consistently. The Liouville equation for the averaged density matrix is isomorphic to a linearized Boltzmann equation, and the effective dephasing rate represents a generalized Bhatnagar-Gross-Krook strong-collision operator. The EDA is applied to the calculation of the ac conductivity of a particle governed by a tight-binding Hamiltonian with static diagonal disorder (the Anderson model). Our results agree with the predictions of scaling theories of the Anderson transition.