Levy flights and multifractality in quantum critical diffusion and in classical random walks on fractals

TL;DR

This paper explores the connection between multifractality, Levy flights, and diffusion in quantum critical systems and classical fractal random walks, revealing common underlying mechanisms through analytical methods.

Contribution

It introduces a virial expansion approach to analyze the density correlation function in critical random matrix ensembles with strong multifractality, linking quantum and classical diffusion behaviors.

Findings

Long-range Hamiltonian effects cause multifractality and Levy flights.

Density correlation functions exhibit power-law behavior at intermediate and long distances.

Quantum and classical models show similar diffusion characteristics.

Abstract

We employed the method of virial expansion in order to compute the retarded density correlation function (generalized diffusion propagator) in the critical random matrix ensemble in the limit of strong multifractality. We found that the long-range nature of the Hamiltonian is a common root of both multifractality and Levy flights which show up in the power-law intermediate- and long-distance behavior, respectively, of the density correlation function. We review certain models of classical random walks on fractals and show the similarity of the density correlation function in them to that for the quantum problem described by the random critical long-range Hamiltonians.