Random Cantor sets and mini-bands in local spectrum of quantum systems

TL;DR

This paper explores the structure of the local spectrum in disordered quantum systems with non-ergodic extended phases, revealing conditions for the emergence of fractal spectra such as random Cantor sets and mini-bands.

Contribution

It introduces a simple model of local density of states with power-law level spacing and derives criteria for fractal spectra based on eigenfunction and spectral fractal dimensions.

Findings

Identification of conditions for singular continuous spectra to form.

Derivation of the correlation function $K()$ and its singularities.

Formulation of a criterion for fractality of the local spectrum.

Abstract

In this paper we give a physically transparent picture of singular-continuous spectrum in disordered systems which possess a non-ergodic extended phase. We present a simple model of identically and independently distributed level spacing in the spectrum of local density of states and show how a fat tail appears in this distribution at the broad distribution of eigenfunction amplitudes. For the model with a power-law local spacing distribution we derive the correlation function $K(\omega)$ of the local density of states and show that depending on the relation between the eigenfunction fractal dimension $D_{2}$ and the spectral fractal dimension $D_{s}$ encoded in the power-law spacing distribution, a singular continuous spectrum of a random Cantor set or that of an isolated mini-band may appear. In the limit of an infinite number of degrees of freedom the function $K(\omega)$ in the non-ergodic extended phase is singular at $\omega=0$ with the branch-cut singularity for the case of a random Cantor set and with the $\delta$-function singularity for the case of an isolated mini-band. For an absolutely continuous spectrum $K(\omega)$ tends to a finite limit as $\omega\rightarrow 0$. For an arbitrary local spacing distribution function we formulated a criterion of fractality of local spectrum and tested it on simple examples.