Interpretation of high-dimensional numerical results for the Anderson transition

Abstract

The existence of the upper critical dimension d c2 = 4 for the Anderson transition is a rigorous consequence of the Bogoliubov theorem on renormalizability of φ4 theory. For d ≥ 4 dimensions, one-parameter scaling does not hold and all existent numerical data should be reinterpreted. These data are exhausted by the results for d = 4, 5 from scaling in quasi-one-dimensional systems and the results for d = 4, 5, 6 from level statistics. All these data are compatible with the theoretical scaling dependences obtained from Vollhardt and Wolfle’s self-consistent theory of localization. The widespread viewpoint that d c2 = ∞ is critically discussed.