Self-Consistent Diagrammatic Theory of Anderson Localization
The model of independent electrons interacting with static, randomly distributed, pointlike scatterers is treated in the extended state perturbation theory at zero temperature. We discuss several types of infrared divergent contributions indicating the onset of Anderson localization. Beyond perturbation theory, using a novel classification scheme of diagrams for the total vertex function, a self-consistent equation for the frequency dependent diffusion coefficient is derived. Solving these equations, one finds that for d ≤ 2 states are always localized, while for d > 2 particles are delocalized for sufficiently weak disorder. We calculate properties in the critical region, e.g. critical exponents and scaling behavior. Defining a frequency dependent conductance we derive a scaling equation and explicitly calculate the corresponding β-function. By a plausible extension we also investigate the length-dependent conductance and its β-function. Where comparison is possible the results agree with those of renormalization group calculations.
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