Unbounded quantum diffusion and fractal spectra

Summary

Abstract

A few years ago we found the existence of an unbounded quantum mechanical diffusion process which strongly contrasted with the dynamical localization of the kicked rotator and other previously studied quantum systems. Here we review how this phenomenon is related to multifractal properties of the spectrum, its level statistics, and to the algebraic decay of correlations. We investigate the influence of classical chaos on uncountable fractal spectra and demonstrate that the concepts of level statistics can be related to multifractal concepts in these cases. First we point out a new class of level statistics where the level spacing distribution follows inverse power laws p(s) ∼ s^−β with 1 < β < 2 and β = 1 + D₀ where D₀ is the fractal dimension of the spectrum. It is characteristic of hierarchical level clustering rather than level repulsion and appears to be universal for systems exhibiting unbounded quantum diffusion where the mean square displacement increases as t^2δ with δ = β − 1. A realization of this class with β = 3/2 is a model of Bloch electrons in a magnetic field (Harper's equation), for which lateral surface superlattices on semiconductor heterojunctions presently serve as experimental realizations. While here diffusion is linear in time (δ = 1/2), in the Fibonacci chain model the spread of wave packets also shows anomalous diffusion (δ ≠ 1/2) with 0 < δ < 1.