Critical Parameters from a Generalized Multifractal Analysis at the Anderson Transition

Abstract

We propose a generalization of multifractal analysis that is applicable to the critical regime of the Anderson localization-delocalization transition. The approach reveals that the behavior of the probability distribution of wave function amplitudes is sufficient to characterize the transition. In combination with finite-size scaling, this formalism permits the critical parameters to be estimated without the need for conductance or other transport measurements. Applying this method to high-precision data for wave function statistics obtained by exact diagonalization of the three-dimensional Anderson model, we estimate the critical exponent ν=1.58±0.03.

Study of off-diagonal disorder using the typical medium dynamical cluster approximation

“…As noted by Anderson, the probability distribution of the local density of states must be considered, focusing on the most probable or the typical value 3,38 . Close to the Anderson transition, the distribution is found to have very long tails characteristic of a log-normal distribution 10,39,40 . In fact, the distribution is log-normal up to ten orders of magnitude 41 and so the typical value 40,[42][43][44] is the geometrical mean.…”

“…Despite progress over the last decades, the subject of Anderson localization remains an active area of research. The lack of quantitative analytical results has meant that numerical investigations [5][6][7][8][9][10][11] have provided a significant role in understanding the Anderson transition 12-14 .…”

Anderson transition and multifractals in the spectrum of the Dirac operator of quantum chromodynamics at high temperature

“…Multifractality is a known feature of critical eigenfunctions at the Anderson metal-insulator transition 6 , that can be studied by means of multifractal finite-size scaling (MFSS) 21 . In recent high-precision calculations [22][23][24] , MFSS has been successfully employed to determine the MFEs of critical eigenfunctions, as well as to obtain a more precise estimate of the critical disorder and of the critical exponents, for Anderson models in different symmetry classes.…”

Anderson universality in a model of disordered phonons

“…The FSS includes corrections to scaling which (i) account for the nonlinearities of the ∆m, ∆k dependence of the scaling variables (relevant scaling) and (ii) for the shift of the point at which the Λ M (ω 2 ) curves cross (irrelevant scaling). This analysis is by now standard and we refer to the literature for details of when fits are acceptable as stable and robust as well as for error estimates via Monte-Carlo approaches [24,25]. Details for the chosen expansions in the present case can be found in Ref.…”