A theory for the conductivity of a fermion gas moving in a strong three-dimensional random potential

An approximation scheme is derived for the dynamical conductivity of a non-interacting fermion gas moving at zero temperature in a three-dimensional random potential, which exhibits a non-linear feedback of the fermion density fluctuation spectrum to the frequency-dependent current relaxation rate. The approximation equations describe an insulator-conductor phase transition caused by strong memory effects and important non-localities in the equations for the current relaxation. Close to the transition point the dynamical conductivity obeys a scaling law with a scaling function which is a solution of an algebraic equation. The main features of the conductivity in the transition regime are a divergence of the insulator polarisability and a continuous decrease of the conductor DC mobility towards a zero at the transition point, a critical slowing down of the current spectrum, a semiconductor-like excitation threshold of the insulator and a strong non-Drudian frequency of the dynamical conductivity.