Quantum Criticality at the Metal Insulator Transition

We introduce a new method to analysis 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 interaction, 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 $T_{SC}\to 0$ giving rise to a metallic phase or $T>T_{SC}$. For repulsive interactions at $T\to 0$ we flow towards the localized phase which is analized within the diffusive Finkelstein theory. For finite temperatures with strong repulsive backward interactions and non-spherical Fermi surfaces characterized by $|\frac{d\ln N(b)}{\ln b}|\ll 1$ one finds a fixed point $(D^*,\Gamma^*_2)$ in the plane $(D,\Gamma_2^{(s)})$. ($D\propto(K_F\ell)^{-1}$ is the disorder coupling constant, $\Gamma_2^{(s)}$ 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 $\rho(D,\Gamma_2^{(s)},T)=\rho(D,\Gamma_2^{(s)},T)\simeq \rho^*f(\frac{D-D^*}{D^*}\frac{1}{T^{z\nu_1}})$ ($\rho^*=\rho(D^*,\Gamma_2^*,1)$, $z=1$, and $\nu_1>1$) in good agreement with the experiments.