Multifractal Analysis with the Probability Density Function at the Three-Dimensional Anderson Transition

Abstract

The probability density function (PDF) for critical wave function amplitudes is studied in the three-dimensional Anderson model. We present a formal expression between the PDF and the multifractal spectrum f(alpha) in which the role of finite-size corrections is properly analyzed. We show the non-Gaussian nature and the existence of a symmetry relation in the PDF. From the PDF, we extract information about f(alpha) at criticality such as the presence of negative fractal dimensions and the possible existence of termination points. A PDF-based multifractal analysis is shown to be a valid alternative to the standard approach based on the scaling of inverse participation ratios.

Distribution of the local density of states as a criterion for Anderson localization: Numerically exact results for various lattices in two and three dimensions

“…To extract meaningful results from numerical studies the data clearly need to be supplemented by a careful finite-size analysis. In this way, the scaling properties of the distribution of the squared wave function amplitude were recently used to determine the multifractal spectrum at the Anderson transition 31 . While in principle all local quantities are equally suited to describe localization properties in terms of their distribution, for practical use the LDOS seems to be the most favorable.…”

Scaling law and critical exponent for α0at the 3D Anderson transition

“…An alternative approach is a direct study of the multifractal spectrum of the wavefunction at the MIT [5][6][7][8]. In principle, the system size dependence of different characteristics of the multifractal spectrum f (α) can be used for such a scaling study.…”

“…In this work, we will determine ν using the parameter α 0 as the scaling variable. α 0 is the location of the maxima of both the multifractal spectrum and the probability density function (PDF) of the variable α = −ln |ψ i | 2 /ln L [8]. Hence ν will be estimated from the raw statistics of wave function intensities |ψ i | 2 .…”

“…This relation between P L (α) and f (α) gives a direct correspondence between the two. Here, the PDF was numerically determined by the histogram of α values [8]. To avoid the effects of bin size caused by the histogram method, the maximum of the PDF is instead more precisely estimated from the inflection point of the cumulative distribution function (CDF).…”

“…To avoid the effects of bin size caused by the histogram method, the maximum of the PDF is instead more precisely estimated from the inflection point of the cumulative distribution function (CDF). Since in the first approximation the P L (α) near the maximum can be regarded to be a Gaussian distribution [8,11], we fit the CDF with [1 + Erf(α − α 0 / √ 2b)]/2 by taking 10% of the total points around its inflection point (α 0 ). The scaling plot as obtained by the current method is shown in Fig.…”