Description of phase transitions

Phenomenological approach

Some basic information concerning theory of the Anderson metal–insulator transition has been given in Chapters 3 and 5. The agreement of the one-parameter scaling hypothesis of Abrahams et al. (1979) with the results of the renormalization group treatment of the nonlinear replica and supermatrix σ-models in 2 + ∈ dimensions was considered by many researchers final proof that the transition was a conventional second-order transition. The only thing that remained to be done was to compute critical exponents, and that could be done by making an expansion in ∈ and putting ∈ = 1 at the end. Other approximate schemes (Götze (1981, 1985), Vollhardt and Wölfle (1980, 1992)) lead to similar results. Although agreement between the exponents computed analytically and those extracted from numerical simulations or experiments was not always good, the validity of the one-parameter scaling description was not usually questioned.

Of course, on the basis of what is known about mesoscopic systems one cannot speak of the average conductance of a finite system, and, possibly, the entire distribution function of the conductances should be scaled. However, renormalization group treatment of the σ-model in 2 + ∈ dimensions does not lead to such a scenario. If one accepts that this approach is appropriate for studying the Anderson transition, the conclusion that the transition is a conventional second-order phase transition is inevitable.