Critical properties of the Anderson localization transition and the high-dimensional limit

In this paper we present a thorough study of transport, spectral, and wave-function properties at the Anderson localization critical point in spatial dimensions d = 3, 4, 5, 6. Our aim is to analyze the dimensional dependence and to assess the role of the d ->infinity limit provided by Bethe lattices and treelike structures. Our results strongly suggest that the upper critical dimension of Anderson localization is infinite. Furthermore, we find that d(U) = infinity is a much better starting point compared to d(L) = 2 to describe even three-dimensional systems. We find that critical properties and finite-size scaling behavior approach by increasing d those found for Bethe lattices: the critical state becomes an insulator characterized by Poisson statistics and corrections to the thermodynamics limit become logarithmic in the number N of lattice sites. In the conclusion, we present physical consequences of our results, propose connections with the nonergodic delocalized phase suggested for the Anderson model on infinite-dimensional lattices, and discuss perspectives for future research studies.