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

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