A generalized thouless formula as a criterion for Anderson localization in two- and three-dimensional systems

A generalized Thouless formula is proposed as an approximate criterion for Anderson localization in two- and three-dimensional systems. The criterion is exact for ordered systems and in the mean field limit of infinite dimensionality. For random systems it is expected that possible corrections are small. Ample numerical evidence is provided through an exact diagonalization of several large-size Hamiltonians. The predictions of the Thouless formula, whose evaluation only needs the eigenvalues, are critically checked against the inverse participation ratio (IPR) of the corresponding eigenvectors. Included are the Lloyd model (for sufficiently low disorder mobility edges cannot be excluded) and surprising, new results on the binary alloy problem. The criterion predicts four as the upper and two as the lower critical dimension.