Local Density of States Correlations in the L\'evy-Rosenzweig-Porter random matrix ensemble

TL;DR

This paper analytically investigates the local density of states correlations in the Le9vy-Rosenzweig-Porter random matrix ensemble, revealing how correlations depend on energy scales and matrix element distributions, supported by numerical validation.

Contribution

It provides the first analytical expressions for local density of states correlations and return probabilities in the Le9vy-Rosenzweig-Porter ensemble, including energy scale dependencies.

Findings

eta( \, ) \, ext{scales as} \, W/\,  ext{ for }  \, ext{much less than} \, _0.

eta( \, ) \, ext{scales as} \, (W/\, ) (/_0)^{-\, } ext{ for }  \, ext{much greater than} \, _0.

ext{Average return probability decays as} \, \, ext{exp}(-(_0 t)^{/2}) ext{ at long times.}

Abstract

We present an analytical calculation of the local density of states correlation function $ \beta(\omega) $ in the L\'evy-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.