Quantum criticality at the metal-insulator transition

We introduce an alternative method to analyze the many-body problem with disorder. The method is an extension of the real space renormalization group based on the operator product expansion. We consider the problem in the presence of interactions, a large elastic mean free path, and finite temperatures. As a result scaling is stopped either by temperature or the length scale set by the diverging many-body length scale (superconductivity). Due to disorder a superconducting instability might take place at T-SC --> 0, giving rise to a metallic phase or T>T-SC. For repulsive interactions at T --> 0 we flow towards the localized phase, which is analyzed within the diffusive Finkelstein theory. For strong repulsive backward interactions and nonspherical Fermi surfaces characterized by \d ln N(b)/ln b\much less than 1 one finds a fixed point (D*, Gamma (2)*) in the plane (D, Gamma ((Delta))(2)). [D proportional to (K (F)iota)(-1) is the disorder coupling constant, Gamma ((Delta))(2) is the particle-hole triplet interaction, b is the length scale, and N(b) is the number of channels.] For weak disorder, D < D*, one obtains a metallic behavior with the resistance <rho>(D, Gamma ((s))(2), T) = rho (D, Gamma ((s))(2), T) similar or equal to rho *f ((D - D*/D* (1/T-z nu1) [rho* = rho (D*, Gamma (2)*, 1), z = 1, and nu (1) > 2], and large ferromagnetic fluctuations caused by the stable fixed point Gamma (2)*.