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Physical review. B, Condensed matter

Critical exponent and multifractality of states at the Anderson transition

Bernhard Krämer

Published Apr 15, 1993• 15 citations
Publisher

Abstract

Based on the assumptions of the existence of a one-paramter scaling law for a self-averaging scaling variable, and on a finite correlation length for the randomness, the lower bound for the critical exponent of the Anderson transition derived earlier by Chayes et al. [Phys. Rev. Lett. 57, 2999 (1986)] for uncorrelated disorder is generalized to the case of spatially correlated disorder. The bound is determined by the system-size dependence of the number of the random variables that are necessary to describe the transition. It is speculated how to relate the critical exponent to the multifractal properties of the states near the critical point. Exponents of several models are estimated and compared with numerical scaling calculations.

DOI
10.1103/physrevb.47.9888

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Based on the assumptions of the existence of a one-paramter scaling law for a self-averaging scaling variable, and on a finite correlation length for the randomness, the lower bound for the critical exponent of the Anderson transition derived earlier by Chayes et al. [Phys. Rev. Lett. 57, 2999 (1986)] for uncorrelated disorder is generalized to the case of spatially correlated disorder. The bound is determined by the system-size dependence of the number of the random variables that are necessary to describe the transition. It is speculated how to relate the critical exponent to the multifractal properties of the states near the critical point. Exponents of several models are estimated and compared with numerical scaling calculations.
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We describe a new multifractal finite-size scaling (MFSS) procedure and its application to the Anderson localization-delocalization transition. MFSS permits the simultaneous estimation of the critical parameters and the multifractal exponents. Simulations of system sizes up to L3=1203{L}^{3}={120}^{3}L3=1203 and involving nearly 106{10}^{6}106 independent wave functions have yielded unprecedented precision for the critical disorder Wc=16.530(16.524,16.536){W}_{c}=16.530(16.524,16.536)Wc​=16.530(16.524,16.536) and the critical exponent $\ensuremath{\nu}=1.590(1.579,1.602)$. We find that the multifractal exponents ${\ensuremath{\Delta}}_{q}$ exhibit a previously predicted symmetry relation and we confirm the nonparabolic nature of their spectrum. We explain in detail the MFSS procedure first introduced in our Letter [Phys. Rev. Lett. 105, 046403 (2010)] and, in addition, we show how to take account of correlations in the simulation data. The MFSS procedure is applicable to any continuous phase transition exhibiting multifractal fluctuations in the vicinity of the critical point.
The multifractal analysis of disorder-induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly-distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent β and the correlation length exponent ν is investigated, We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent ν are obtained by the typical order parameter exponent α0 and make contact between the multifractal analysis and the finite size scaling approach in 2-d by relying on conformal mapping arguments.
The multifractal analysis of disorder induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent β and the correlation length exponent ν is investigated. We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent ν are obtained by the typical order parameter exponent α0 and make contact between the multifractal analysis and the finite size scaling approach in 2-d by relying on conformal mapping arguments. 1.
We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wave-function intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4\ifmmode\times\else\texttimes\fi{}{10}^{6}$ independently generated wave functions of system sizes up to L3=1503{L}^{3}={150}^{3}L3=1503, yields accurate estimates for the critical exponent of the localization length, $\ensuremath{\nu}=1.446(1.440,1.452)$, the critical value of the disorder strength, and the multifractal exponents.
Based on finite-size scaling ideas a new approach to analyse the singularities associated with second-order phase transitions is presented. Under certain requirements, the method allows to derive an upper limit for the critical exponent from finite-size data that describe the system far away from the transition. By extrapolation the critical exponent ν is determined. As an example, the approach is applied to the metal-insulator transition in disordered systems. The result ν = 1.35 ± 0.15 is independent of the employed random distribution. With field theoretical arguments we discuss and generalize this result. Our results are consistent with the assumption that one-parameter scaling and universality are valid in the 3D Anderson model.