Self-consistent mode-coupling theory of quantum percolation

We present a novel treatment of the quantum percolation problem that yields a hierarchy of self-consistent equations for the frequency-dependent diffusion coefficient. The first member of this hierarchy is derived and analyzed. This equation predicts that excitations are localized in the presence of any disorder in one and two dimensions, but that a transition from localized to extended excitations takes place for a disordered three-dimensional system. The predicted critical behavior is identical to that obtained by Vollhardt and W\"olfle for a Fermi gas in a random potential. The frequency dependence of the ac conductivity and the critical exponents determined from the self-consistent equation agree with the predictions of the scaling theory of Anderson localization.