Density of states correlations in L\'evy Rosenzweig-Porter model via supersymmetry approach

TL;DR

This paper calculates the density-of-states correlation function in the Le9vy-Rosenzweig-Porter model using supersymmetry, revealing oscillatory behavior at low energies and matching cavity approach results at high energies.

Contribution

It extends Efetov's supersymmetry method to exactly compute the density-of-states correlations in the Le9vy-Rosenzweig-Porter ensemble across all energy scales.

Findings

GUE-type oscillations at low energies with exponential decay

Matching results with cavity approach at high energies

Inverse miniband width related to local decay times distribution

Abstract

We studied global density-of-states correlation function $R(\omega)$ for L\'evy-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 $\omega \leq \Gamma$\, (with $\Gamma \gg \Delta$ being the effective miniband width) we found GUE-type oscillations with period of level spacing $\Delta$, decaying exponentially at the Thouless energy scale $E_{Th} = \sqrt{\Delta \Gamma/2\pi}$. At high energies $\omega \gg E_{Th}$ our results coincide with those obtainen via cavity equation approach. Inverse of the effective miniband width, $1/\Gamma$, is shown to be given by the average of the local decay times over L\'evy distribution.