Finite-size scaling approach to Anderson localisation. II. Quantitative analysis and new results

Abstract

For pt.I see ibid., vol.14, p.L127 (1981). In three dimensions, there is a transition at a critical value W_c of the disorder parameter from a region of exponentially localised states to a region of extended states. When W decreases to W_c, the authors find a value equal to 0.66 for the critical exponent nu relative to the divergence of the localisation length. On the other side of the transition, they are naturally led to define an 'order parameter' and give its variation against W in the whole region of extended states. In two dimensions, a critical value W_c of the disorder parameter separates a region of exponentially localised states (W>W_c) from a region of 'quasi-extended' states which are non-square summable and fall off as 1/R^{eta (W)}. They give the variation of the exponent eta (W) in the whole 'quasi-extended' region. On the other hand, they show that the divergence of the localisation length when W decreases to W_c is now controlled by an essential singularity. As a conclusion, they describe the striking analogies with their results in any dimension and the behaviour of the 'XY' phase transition model.