Laboratoire de Physique Théorique
Toulouse - UMR 5152

It has been known for some time that at the Anderson transition the Kondo temperature T_K has a wide distribution with a power law tail towards low T_K. It was predicted (Kettemann et al., 2012) that the exponent of this power law should be universal and have a value related to the multifractal exponent eta describing wavefunction correlations at the critical point. The analytic calculation involves certain approximations and it seemed worthwhile to check the prediction numerically. In this talk, I will discuss the work reported in Slevin et al. (2019), where the T_K distribution was investigated in numerical simulations of Anderson’s model of localisation.

Large ensembles of three-dimensional cubic samples were simulated for several exchange couplings and systems sizes. The local density of states for each sample was calculated using the kernel polynomial method and T_K for each sample was estimated by solving the Nagaoka-Suhl equation. To compare the distributions of T_K obtained in the simulations with the analytic prediction, the method of Clauset et al was used.

We found a value for the exponent of the power law tail which was, within the precision of the simulations, independent of the exchange coupling and the system size. The value found was slightly below the analytic prediction. We speculate that this may relate to the approximation of a parabolic multifractal spectrum in the analytic work. However, the numbers of free moments found greatly exceed the analytic predictions. For the moment, this remains a puzzle.

S. Kettemann, E. R. Mucciolo, I. Varga and K. Slevin (2012), "Kondo-Anderson transitions." Physical Review B 85(11) : 115112.

K. Slevin, S. Kettemann and T. Ohtsuki (2019), "Multifractality and the distribution of the Kondo temperature at the Anderson transition." The European Physical Journal B92(12) : 281.

A. Clauset, C. R. Shalizi and M. E. J. Newman (2009), "Power-Law Distributions in Empirical Data." SIAM Review 51(4) : 661-703.