Local Density of States Correlations in the Lévy-Rosenzweig-Porter random matrix ensemble

Abstract

We present an analytical calculation of the local density of states correlation function $\beta (\omega )$ in the Lévy-Rosenzweig-Porter random matrix ensemble at energy scales larger than the level spacing but smaller than the bandwidth. The only relevant energy scale in this limit is the typical level width ${\Gamma }_0$. We show that $\beta (\omega \ll {\Gamma }_0)\sim W/{\Gamma }_0$ (here $W$ is width of the band) whereas $\beta (\omega \gg {\Gamma }_0)\sim (W/{\Gamma }_0)(\omega /{\Gamma }_0{)}^{-\mu }$ where $\mu$ is an index characterising the distribution of the matrix elements. We also provide an expression for the average return probability at long times: $\ln [R(t\gg {\Gamma }_0^{-1})]\sim -({\Gamma }_{0}t{)}^{\mu /2}$. Numerical results based on the pool method and exact diagonalization are also provided and are in agreement with the analytical theory.

arXiv comment19 pages, 3 figures

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• Free Probability approach to spectral and operator statistics in Rosenzweig-Porter random matrix ensembles
• The Wishart--Rosenzweig--Porter random matrix ensemble
• Density of states correlations in Lévy Rosenzweig-Porter model via supersymmetry approach