Symmetry Theory of the Anderson Transition

Abstract

We prove the Vollhardt and Wolfle hypothesis that the irreducible vertex U_{kk'}(q) appearing in the Bethe--Salpeter equation contains a diffusion pole (with the observable diffusion coefficient D(\omega,q)) in the limit k+k'\to 0. The presence of a diffusion pole in U_{kk'}(q) makes it possible to represent the quantum "collision operator" L as a sum of a singular operator L_{sing}, which has an infinite number of zero modes, and a regular operator L_{reg} of a general form. Investigation of the response of the system to a change in L_{reg} leads to a self-consistency equation, which replaces the rough Vollhardt-Wolfle equation. Its solution shows that D(0,q) vanishes at the transition point simultaneously for all q. The spatial dispersion of D(\omega,q) at \omega \to 0 is found to be \sim 1 in relative units. It is determined by the atomic scale, and it has no manifestations on the scale q \sim \xi^{-1} associated with the correlation length \xi. The values obtained for the critical exponent s of the conductivity and the critical exponent \nu of the localization length in a d-dimensional space, s=1 (d>2) and \nu=1/(d-2) (2<d<4), \nu=1/2 (d>4), agree with all reliably established results. With respect to the character of the change in the symmetry, the Anderson transition is found to be similar to the Curie point of an isotropic ferromagnet with an infinite number of components. For such a magnet, the critical exponents are known exactly and they agree with the exponents indicated above. This suggests that the symmetry of the critical point has been established correctly and that the exponents have been determined exactly.