One-parameter scaling: some open questions

Abstract

After a re-statement of the one-parameter scaling ansatz in the strong (global) sense, the main predictions of and the experimental/numerical support for it are briefly reviewed. A mathematical condition implied by this ansatz for the simplest model with a white-noise Gaussian potential characterized by the disorder parameter k_Fle and the sample size Lll_e is pointed out. It is shown that in the 1D case the one-parameter scaling in terms of distribution of non-selfaveraging resistance holds only in the limit of weak disorder and long sample length. The critical role of coherent-backscattering is discussed and it is suggested that its suppression by a strong magnetic field may not eliminate the mobility-edge, but may alter the nature of the transition qualitatively. The possible role of resonances interpolating between the strong and the weak localization regimes is pointed out. It is argued that in a magnetic field the non-perturbative physics at the mobility edge is similar to that at the ionization threshold of a hydrogenic potential problem and may reveal the fine-structure of the mobility edge.