Mathematical characterization and shape analysis of localized, fractal, and complex distributions in extended systems

Abstract

The availability of recent supercomputers and massively parallel computing facilities makes possible the calculation of the electronic structure of highly extended (rnesoscopic) molecular networks. Disorder, which is practically always present in these systems, causes an extreme complexity of the wave function that typically shows multifractal behavior in the intermediate length scale. Multifractal analysis, however, is possible only on systems that cover several orders of length scales. Though such calculation can be canied out on model systems, it is beyond the bounds of present ab initio or semiempirical treatments. In this contribution, a shape-analysis method of the wave function is given that is applicable both for localized and multifractal one-particle states even in moderately large networks without a regular geometrical structure. No boxing of the distributions is necessary through several orders of magnitude of scaling distances. Multifractal behavior and different regularly decaying localization shape functions can be distinguished. Finite-size multifractal distributions are also discussed. The described method is intended to serve as an easily applicable and efficient tool for bridging over the gap between the wave-function analysis of systems containing macroscopic and moderately large number of particles.

Rényi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems

“…However, as it has been demonstrated in Ref. 12 and applied in several studies later 12,13,14,15,16 , the difference…”

“…The participation ratio, q(ξ), in the limit ξ → ∞ (L → ∞ for fixed σ or σ → 0 for fixed L) tends to zero as q(ξ) ≈ 2 √ π/ξ while S str (ξ) → S G str . On the other hand in the other limit of ξ → 0 (σ → ∞ for fixed L or L → 0 for fixed σ) we see that q(ξ) ≈ (1 − ξ 4 /720) and S str (ξ) ≈ ξ 4 /1440 therefore the relation S str ≈ (1 − q)/2 is also fulfilled 12 . We would like to emphasize that no divergences are found for parameters q and S str and they show well defined behavior in either limit.…”

Multifractality beyond the parabolic approximation: Deviations from the log-normal distribution at criticality in quantum Hall systems

“…Generalizations for higher moments are also possible [17], however, numerical simulations for such cases are presently not too reliable. The above two parameters have already been successfully applied in a number of systems [12,14,15] for the shape-analysis of the complex distribution function of eigenvector components and energy spacing distributions.…”

The generalized localization lengths in one-dimensional systems with correlated disorder

“…We observe that the RBDM is well described by an overall shape: c m ∼ m −3 (dashed line) while the GRDM and the DKPM are better described with c m ∼ m −6 (dashed-dotted line). Apparently this is in contradiction with analytical expressions of the Lyapunov exponent (inverse localization length) which goes as γ(E) ∼ (E − E c ) 2 around the special energies E c [6,15], however, γ should vanish for the case of power-law localization [23,24]. A possible resolution to this problem is already outlined in Section 2, e.g.…”