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arXiv:cond-mat/0306024 · Physical review. B, Condensed matter

Multifractality of Hamiltonians with power-law transfer terms

Emilio Cuevas

Published Nov 21, 2003• 26 citations
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Abstract

Finite-size effects in the generalized fractal dimensions ${d}_{q}$ are investigated numerically. We concentrate on a one-dimensional disordered model with long-range random hopping amplitudes in both the strong- and the weak-coupling regime. At the macroscopic limit, a linear dependence of ${d}_{q}$ on q is found in both regimes for values of $q\ensuremath{\lesssim}{4g}^{\ensuremath{-}1},$ where g is the coupling constant of the model.

DOI
10.1103/physrevb.68.184206

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Finite-size effects in the generalized fractal dimensions dq{d}_{q}dq​ are investigated numerically. We concentrate on a one-dimensional disordered model with long-range random hopping amplitudes in both the strong- and the weak-coupling regime. At the macroscopic limit, a linear dependence of dq{d}_{q}dq​ on q is found in both regimes for values of q\ensuremath≲4g\ensuremath−1,q\ensuremath{\lesssim}{4g}^{\ensuremath{-}1},q\ensuremath≲4g\ensuremath−1, where g is the coupling constant of the model.
The system size dependence of the multifractal spectrum $f(\ensuremath{\alpha})$ and its singularity strength $\ensuremath{\alpha}$ is investigated numerically. We focus on one-dimensional (1D) and 2D disordered systems with long-range random hopping amplitudes in both the strong and the weak disorder regime. At the macroscopic limit, it is shown that $f(\ensuremath{\alpha})$ is parabolic in the weak disorder regime. In the case of strong disorder, on the other hand, $f(\ensuremath{\alpha})$ strongly deviates from parabolicity. Within our numerical uncertainties it has been found that all corrections to the parabolic form vanish at some finite value of the coupling strength.
Some properties of ddd-dimensional disordered models with long-range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension d2d_2d2​ (for d=2d=2d=2) and the nearest level spacing distribution Pc(s)P_c(s)Pc​(s) (for d=3d=3d=3) in both the weak (bd≫1b^d \gg 1bd≫1) and the strong (bd≪1b^d \ll 1bd≪1) coupling regime, where the parameter b−db^{-d}b−d plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of d2d_2d2​ are of the form d2=cdbdd_2=c_db^dd2​=cd​bd in the strong coupling limit and d2=d−ad/bdd_2=d-a_d/b^dd2​=d−ad​/bd in the case of weak coupling, and (ii) P(s)P_ (s)P(​s) has the asymptotic form Pc(s)∼exp⁡(−Adsα)P_c(s)\sim\exp (-A_ds^{\alpha})Pc​(s)∼exp(−Ad​sα) for $s\gg $, with the critical exponent α=2−ad/bd\alpha=2-a_d/b^dα=2−ad​/bd for bd≫1b^d \gg 1bd≫1 and α=1+cdbd\alpha=1+c_d b^dα=1+cd​bd for bd≪1b^d \ll 1bd≪1. In these cases the numerical coefficients AdA_dAd​, ada_dad​ and cdc_dcd​ depend only on the dimensionality.
Long-range hoppings in quantum disordered systems are known to yield quantum multifractality, the features of which can go beyond the characteristic properties associated with an Anderson transition. Indeed, critical dynamics of long-range quantum systems can exhibit anomalous dynamical behaviors distinct from those at the Anderson transition in finite dimensions. In this paper, we propose a phenomenological model of wave packet expansion in long-range hopping systems. We consider both their multifractal properties and the algebraic fat tails induced by the long-range hoppings. Using this model, we analytically derive the dynamics of moments and inverse participation ratios of the time-evolving wave packets, in connection with the multifractal dimension of the system. To validate our predictions, we perform numerical simulations of a Floquet model that is analogous to the power law random banded matrix ensemble. Unlike the Anderson transition in finite dimensions, the dynamics of such systems cannot be adequately described by a single parameter scaling law that solely depends on time. Instead, it becomes crucial to establish scaling laws involving both the finite size and the time. Explicit scaling laws for the observables under consideration are presented. Our findings are of considerable interest towards applications in the fields of many-body localization and Anderson localization on random graphs, where long-range effects arise due to the inherent topology of the Hilbert space.
Abstract Some properties of d‐dimensional disordered models with long‐range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension d2 (for d = 2) and the nearest level spacing distribution Pc(s) (for d = 3) in both the weak (bd ≫ 1) and the strong (bd ≪ 1) coupling regime, where the parameter b− d plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of d2 are of the form d2 = cdbd in the strong coupling limit and d2 = d − ad/bd in the case of weak coupling, and (ii) Pc(s) has the asymptotic form Pc(s) ∼ exp (−Adsα) for s ≫ 1, with the critical exponent α = 2 − ad/bd for bd ≫ 1 and α = 1 + cdbd for bd ≪ 1. In these cases the numerical coefficients Ad, ad and cd depend only on the dimensionality. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We investigate the wave-packet dynamics of the power-law bond disordered one-dimensional Anderson model with hopping amplitudes decreasing as ${H}_{nm}\ensuremath{\propto}{\ensuremath{\mid}n\ensuremath{-}m\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}}$. We consider the critical case $(\ensuremath{\alpha}=1)$. Using an exact diagonalization scheme on finite chains, we compute the participation moments of all stationary energy eigenstates as well as the spreading of an initially localized wave packet. The eigenstates multifractality is characterized by the set of fractal dimensions of the participation moments. The wave packet shows a diffusivelike spread developing a power-law tail and achieves a stationary nonuniform profile after reflecting at the chain boundaries. As a consequence, the time-dependent participation moments exhibit two distinct scaling regimes. We formulate a finite-size scaling hypothesis for the participation moments relating their scaling exponents to the ones governing the return probability and wave-function power-law decays.
We consider a simple model of quantum disorder in two dimensions, characterized by a long-range site-to-site hopping. The system undergoes a metal–insulator transition--its eigenfunctions change from being extended to being localized. We demonstrate that at the point of the transition the nature of the eigenfunctions depends crucially on the magnitude of the hopping amplitude. At small amplitudes they are strongly multifractal. In the opposite limit of large amplitudes, the eigenfunctions do not become fractal. Their density moments do not scale as a power of the system size; instead our result suggests a power of the logarithm of the system size. In this regard, the transition differs from a similar one in the one-dimensional version of the same system, as well as from the conventional Anderson transition in more than two dimensions.