Dynamical Scaling Properties of Electrons in Quantum Systems with Multifractal Eigenstates

TL;DR

This paper investigates how the dynamical scaling of electron wave packets relates to the multifractal nature of eigenstates in quantum systems, revealing a proportionality between the scaling exponent and the correlation dimension.

Contribution

It establishes a quantitative relationship between wave packet dynamics and eigenstate multifractality, extending previous understanding to higher-dimensional systems and different motion regimes.

Findings

The root mean square displacement scales as t^β with β=D_2^ψ in certain quantum models.

The relationship β=D_2^ψ/d holds in d-dimensional systems with ballistic motion.

Non-ballistic motion leads to β<D_2^ψ/d, supported by numerical evidence.

Abstract

We study the intricate relationships between the dynamical scaling properties of electron wave packets and the multifractality of the eigenstates in quantum systems. Numerical simulations for the Harper model and the Fibonacci chain indicate that the root mean square displacement displays the scaling behavior $r(t)\sim t^\beta$ with $\beta=D_2^\psi$, where $D_2^\psi$ is the correlation dimension of the multifractal eigenstates. The equality can be generalized to $d$-dimensional systems as $\beta=D_2^\psi/d$, as long as the electron motion is ballistic in the effective $D_2^\psi$-dimensional space. This equality should be replaced by $\beta<D_2^\psi/d$ if the motion is non-ballistic, as supported by all known results.