Relation between the Correlation Dimensions of Multifractal Wavefunctions and Spectral Measures in Integer Quantum Hall Systems

TL;DR

This study links the decay of wavepacket return probability in quantum Hall systems to the multifractal nature of spectral measures, revealing a novel relationship between wavefunction and spectral dimensions at the metal-insulator transition.

Contribution

It demonstrates that the decay exponent of wavepacket return probability equals the spectral measure's generalized dimension, establishing a direct connection between wavefunction and spectral multifractality.

Findings

Return probability decays algebraically with exponent D_2/2

Spectral measure is multifractal

Exponent D_2/2 equals spectral measure's generalized dimension

Abstract

We study the time evolution of wavepackets of non-interacting electrons in a two-dimensional disordered system in strong magnetic field. For wavepackets built from states near the metal-insulator transition in the center of the lowest Landau band we find that the return probability to the origin $p(t)$ decays algebraically, $p(t) \sim t^{-D_2/2}$, with a non-conventional exponent $D_2/2$. $D_2$ is the generalized dimension describing the scaling of the second moment of the wavefunction. We show that the corresponding spectral measure is multifractal and that the exponent $D_2/2$ equals the generalized dimension $\widetilde{D}_2$ of the spectral measure.