Spatial and Spectral Multifractality of the Local Density of States at the Mobility Edge

TL;DR

This study numerically investigates the multifractal properties of the local density of states at the mobility edge in disordered systems, revealing proportionality between spatial and spectral multifractal dimensions.

Contribution

It introduces a numerical approach to analyze the multifractality of LDOS at the mobility edge, establishing a proportionality relation between spatial and spectral measures.

Findings

Spatial and spectral multifractal dimensions are proportional at the mobility edge.

The energy dependence of LDOS can be efficiently obtained from wavefunction time evolution.

The proportionality relation supports the effective system size concept based on frequency-dependent length scale.

Abstract

We performed numerical calculations of the local density of states (LDOS) at disorder induced localization-delocalization transitions. The LDOS defines a spatial measure for fixed energy and a spectral measure for fixed position. At the mobility edge both measures are multifractal and their generalized dimensions $D(q)$ and $\tilde{D}(q)$ are found to be proportional: $D(q)=d\tilde{D}(q)$, where $d$ is the dimension of the system. This observation is consistent with the identification of the frequency-dependent length scale $L_\omega \propto \omega^{-1/d}$ as an effective system size. The calculations are performed for two- and three-dimensional dynamical network models with local time evolution operators. The energy dependence of the LDOS is obtained from the time evolution of the local wavefunction amplitude of a wave packet, providing a numerically efficient way to obtain information about the multifractal exponents of the system.