Density of states correlations in Lévy Rosenzweig-Porter model via supersymmetry approach

We studied global density-of-states correlation function $R(\omega )$ for Lévy-Rosenzweig-Porter random matrix ensemble in the non-ergodic extended phase. Using an extension of Efetov’s supersymmetry approach we calculated $R(\omega )$ exactly in all relevant ranges of $\omega$. At relatively low $\omega \le \mathrm{\Gamma }$, (with $\mathrm{\Gamma }\gg \mathrm{\Delta }$ being the effective miniband width) we found GUE-type oscillations with period of level spacing $\mathrm{\Delta }$, decaying exponentially at the Thouless energy scale $E_{Th}=\sqrt{\mathrm{\Delta }\mathrm{\Gamma }\mathrm{/}2\pi }$. At high energies $\omega \gg E_{Th}$ our results coincide with those obtainen via cavity equation approach. Inverse of the effective miniband width, $1\mathrm{/}\mathrm{\Gamma }$, is shown to be given by the average of the local decay times over Lévy distribution.