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 $ β(ω) $ 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 $Γ_0$. We show that $β(ω\ll Γ_0) \sim W/Γ_0$ (here $W$ is width of the band) whereas $β(ω\gg Γ_0) \sim (W/Γ_0) (ω/Γ_0)^{-μ} $ where $μ$ is an index characterising the distribution of the matrix elements. We also provide an expression for the average return probability at long times: $\ln [R(t\ggΓ_0^{-1})] \sim -(Γ_0 t)^{μ/2}$. Numerical results based on the pool method and exact diagonalization are also provided and are in agreement with the analytical theory.