We consider a model of $N$ species of electrons in a random potential interacting via a short-range repulsive interaction. We study the $N=\infty$ limit and the $\frac{1}{N}$ expansion to the leading order in $\frac{1}{N}$. 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 $2+\epsilon$ 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 $\frac{1}{N}$, (a) the exponent $\nu$ of the localization length is the same as in the noninteracting theory, (b) the dc conductivity vanishes at the mobility edge with an exponent $s=\frac{2}{17}$, (c) the density of states at the Fermi surface vanishes at the mobility edge with an exponent $\delta =\frac{2}{17}$, (d) the mean free time $\tau$ at the Fermi surface vanishes at the mobility edge with an exponent $\zeta =\frac{7}{17}$, (e) the Fermi velocity diverges at the mobility edge with an exponent $\rho =\frac{5}{17}$, and (f) the diffusive modes acquire wave-function renormalization and the anomalous dimension $\eta$ is (to leading order) equal to $\frac{\epsilon }{34}$.

• Received 24 March 1983

DOI:https://doi.org/10.1103/PhysRevB.28.2990