Interpretation of high-dimensional numerical results for Anderson transition

TL;DR

This paper discusses the critical dimension for the Anderson transition, showing that for dimensions four and above, traditional one-parameter scaling fails and existing numerical data align with self-consistent localization theory.

Contribution

It clarifies the upper critical dimension for Anderson transition as four, reinterprets existing numerical data, and challenges the view that the critical dimension is infinite.

Findings

Critical dimension d_{c2} = 4 confirmed for Anderson transition.

Numerical data for d=4,5,6 are consistent with self-consistent localization theory.

One-parameter scaling does not hold for d ≥ 4.

Abstract

Existence of the upper critical dimension d_{c2}=4 for the Anderson transition is a rigorous consequence of the Bogoliubov theorem on renormalizability of \phi^4 theory. For dimensions d\ge 4, one-parameter scaling does not hold, and all existent numerical data should be reinterpreted. These data are exhausted by results for d=4,5 from scaling in quasi-one-dimensional systems, and results for d=4,5,6 from level statistics. All these data are compatible with the theoretical scaling dependencies obtained from self-consistent theory of localization by Vollhardt and Woelfle. The critical discussion is given for a widespread point of view that d_{c2}=\infty.