The Critical Exponents of the 2D and 3D Anderson Transitions

Abstract

Since 1999, the year of the last Localisation conference in Hamburg, there have been several key advances in the numerical analysis of the Anderson transition. A working method for taking account of corrections to scaling when analysing numerical data has been demonstrated with an accompanying improvement in the accuracy of the estimate of the critical exponent. 1) Second, single parameter scaling of the conductance distribution of a three dimensional mesoscopic conductor in the critical regime has been demonstrated. 2,3) This has shown how the phenomenon of mesoscopic conductance fluctuations and the scaling theory of localisation may be reconciled, as well as providing independent numerical estimates of the critical exponent. These advances are welcome, especially as it has proved impossible to estimate the critical exponent reliably with any of the usual analytic approaches such as the � -expansion. Here we report estimates of the critical exponent ν, which describes the divergence of the localisation length, for the Anderson transitions in the orthogonal universality class in three dimensions (3D) and the symplectic universality class in two dimensions (2D). Symplectic symmetry is realised in systems with time reversal symmetry but where spin-rotation symmetry is broken by the spin-orbit interaction. In this case, by simulating an SU(2) model for which corrections to scaling are negligible, we have been able to estimate the exponent far more accurately than previously possible. 4) 2. 3D orthogonal university class