Localization and interactions in a disordered electron gas

We consider a model of N species of electrons in a random potential interacting via a short-range repulsive interaction. We study the N=∞ limit and the 1/N expansion to the leading order in 1/N. After renormalizing the theory, we find that there are three coupling constants in this problem: (i) a coupling constant with the dimensions of the resistivity, (ii) the coupling for electron-electron scattering, and (iii) the coupling strength between diffusive modes and density fluctuations. The renormalization-group equations are presented. In 2+ε dimensions the Anderson fixed point of the noninteracting theory is shown to belong to a line of unstable fixed points. A new ("interacting") fixed point is found. At the transition we find that, to leading order in 1/N, (a) the exponent ν of the localization length is the same as in the noninteracting theory, (b) the dc conductivity vanishes at the mobility edge with an exponent s=2/17, (c) the density of states at the Fermi surface vanishes at the mobility edge with an exponent δ=2/17, (d) the mean free time τ at the Fermi surface vanishes at the mobility edge with an exponent ζ=7/17, (e) the Fermi velocity diverges at the mobility edge with an exponent ρ=5/17, and (f) the diffusive modes acquire wave-function renormalization and the anomalous dimension η is (to leading order) equal to ε34.