On the range of validity of a semiclassical relation between the critical exponents at the Anderson localization transition

Abstract

Using Wegner's result relating critical exponents s and ν for conductivity and localization length, respectively, via dimensionality d and that for ν given by García-García, we derive what we term a semiclassical (sc) relation for ν in terms of s, which is independent of dimensionality. Forming the ratio s/ ν versus d from the above relations, s/ ν=0 at d=2 is due to a singularity in the sc relation for ν. We argue that, in reality, s/ ν=0 results from s being zero at d=2. Finally we conjecture that (i) Wegner's prediction s/ ν=1 when d=3 and (ii) ν tends to 1/2 at large s, are both insensitive to interactions.