Dynamical Scaling Properties of Electrons in Quantum Systems with Multifractal Eigenstates

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.