Multifractal analysis of the metal-insulator transition in the three-dimensional Anderson model. II. Symmetry relation under ensemble averaging

Abstract

We study the multifractal analysis ͑MFA͒ of electronic wave functions at the localization-delocalization transition in the three-dimensional Anderson model for very large system sizes up to 240 3 . The singularity spectrum f͑␣͒ is numerically obtained using the ensemble average of the scaling law for the generalized inverse participation ratios P q , employing box-size and system-size scaling. The validity of a recently reported symmetry law ͓Mirlin et al., Phys. Rev. Lett. 97, 046803 ͑2006͔͒ for the multifractal spectrum is carefully analyzed at the metal-insulator transition. The results are compared to those obtained using different approaches, in particular the typical average of the scaling law. System-size scaling with ensemble average appears as the most adequate method to carry out the numerical MFA.

Typical medium dynamical cluster approximation for the study of Anderson localization in three dimensions

“…W c is defined by the vanishing of the TDOS (ω = 0). Our results show that as cluster size N c increases, for N c 12 the W c systematically increases until it converges to W c ≈ 2.1 (for details, see Supplemental Material (SM) [26]) which is in good agreement with the values reported in the literature [4,17,18,[20][21][22]27]. This cluster is the first one with a complete NN shell (for details, see SM [26]).…”

“…It underestimates the critical disorder strength with W TMT c ≈ 1.65 instead of the numerical value W c ≈ 2.1 [17][18][19][20] (in a unit where 4t = 1), and the critical exponent of the order parameter β TMT ≈ 1.0, whereas the recently reported value is β ≈ 1.67 [20,21]. Another crucial drawback of the local TMT in 3D is that it cannot describe the reentrant behavior of the mobility edge (the energy separating extended and localized electron states) as seen in transfer matrix method studies [4,22].…”

“…From this cluster onward, W c converges to ≈ 2.1. Fitting the data for the two largest clusters starting from W ≈ 1.0 with a power law TDOS (ω = 0) = a 0 |W − C| β , we obtain β > 1.40, which is greater than a single site TMT value of β T MT = 1.0; but it is still smaller than the most recently reported β ≈ 1.67 [20,21,28]. We note that other mean-field methods reported β 1.0 [29].…”

“…We note that other mean-field methods reported β 1.0 [29]. In our method, we note that it is unlikely that we can calculate the critical disorder strength and exponents as precisely as diagonalization and transfer matrix methods [4,17,18,[20][21][22]27]. However, the advantage of our method is that we can incorporate both interactions and realistic electronic structure as in, e.g., the dynamical mean-field theory [30] and other DCA calculations (see, e.g., Ref.…”

“…This is due to the fact that the "typical" value of a "random" variable corresponds to the most probable value of the PDF [21,24].…”