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High precision multifractal analysis in the 3D Anderson model of localisation

By Louella J. Vasquez


This work presents a large scale multifractal analysis of the electronic state\ud in the vicinity of the localisation-delocalisation transition in the three-dimensional\ud Anderson model of localisation using high-precision data and very large system\ud sizes of up to L3 = 2403. The multifractal analysis is implemented using box- and\ud system- size scaling of the generalized inverse participation ratios employing typical\ud and ensemble averaging techniques. The statistical analysis in this study has shown\ud that in the thermodynamic limit a proposed symmetry relation in the multifractal\ud exponents is true for the 3D Anderson model in the orthogonal universality class.\ud Better agreement with the symmetry is found when using system-size scaling with\ud ensemble averaging in which a more complete picture of the multifractal spectrum\ud f(α) is also obtained. A complete profile of f(α) has negative fractal dimensions\ud and shows the contributions coming from the tails of the distribution. Various boxpartitioning\ud approaches have been carefully studied such as the use of cubic and\ud non-cubic boxes, periodic boundary conditions to enlarge the system, and single\ud and multiple origins in the partitioning grid. The most reliable method is equal\ud partitioning of a system into cubic boxes which has also been shown to be the least\ud numerically expensive. Furthermore, this work gives an expression relating f(α)\ud and the probability density function (PDF) of wavefunction intensities. The relation\ud which contains a finite-size correction provides an alternative and simpler method\ud to obtain f(α) directly from the PDF in which f(α) is interpreted as the scaleinvariant\ud distribution at criticality. Finally, a generalization of standard multifractal\ud analysis which is applicable to the critical regime and not just at the critical point is\ud presented here. Using this generalization together with finite-size scaling analysis,\ud estimates of critical disorder and critical exponent based on exact diagonalization\ud have been obtained that are in excellent agreement, supporting for the first time\ud previous results of transfer matrix calculations

Topics: QA, QC
OAI identifier: oai:wrap.warwick.ac.uk:3811

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