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

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 $\omega \to 0$. For an arbitrary local spacing distribution function we formulated a criterion of fractality of local spectrum and tested it on simple examples.

38 pages, 10 figures, dedicated to memory of K. B. Efetov

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Cited by in corpus (10)

• Absence of Mobility Edge in Short-range Uncorrelated Disordered Model: Coexistence of Localized and Extended States
• Tuning the phase diagram of a Rosenzweig-Porter model with fractal disorder
• Robust non-ergodicity of ground state in the $\beta$ ensemble
• Anatomy of the eigenstates distribution: a quest for a genuine multifractality
• Spectral form factor and energy correlations in banded random matrices
• Re-entrant localization induced by short-range hopping in the fractal Rosenzweig-Porter Model
• Quantum logarithmic multifractality
• Anomalous energy correlations and spectral form factor in the nonergodic phase of the $\beta$-ensemble
• Universal Relation between Spectral and Wavefunction Properties at Criticality
• Ergodicity-breaking phase diagram and fractal dimensions in long-range models with generically correlated disorder