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

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.