Scaling Exponents at a Mobility Edge in Two Dimensions
It is now widely recognized that electrons in very dirty metals localize  and an Anderson metal-insulator transition occurs in disordered lattice systems when the Fermi level crosses a mobility edge which separates extended from localized states in the energy spectra. Such viewpoint of localization relies on a simplified non-interacting electron picture. In this context a one- parameter scaling theory  offers the theoretical framework to understand the results for the associated critical behavior within a few universality classes. In fact, three universality classes are distinguished , depending on symmetry: the orthogonal in the case of a random potential, the unitary when a magnetic field is added and the symplectic when spin-orbit coupling is also present. A firm prediction exists for the presence of a phase transition for the orthogonal universality class only in three or higher dimensions. Two-dimensional (2D) disordered systems, however, are believed to display mobility edges and extended states when spin-orbit coupling or a strong magnetic field are present, that is for the symplectic and unitary universality classes, respectively. Our theme is precisely the study of a mobility edge in 2 D and the details of the associated critical behaviour which are largely open questions. 2 D is also an especially promising area to study conformai invariance . Emphasis is placed on the following aspects: Firstly, the evaluation of the localization length exponent v and the correlation exponent by scaling the dominant Lyapunov exponent near the transition. Secondly, a complete numerical analysis of the spectral distributions for all the Lyapunov exponents in connection with the predictions of weak disorder theories.
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