Spectral properties of Levy Rosenzweig-Porter model via supersymmetric approach

TL;DR

This work addresses the spectral statistics of random matrices with heavy-tailed off-diagonal elements (Lévy and Lévy-Rosenzweig-Porter models) and their ergodic-to-non-ergodic transitions. It deploys Efetov's supersymmetric formalism with a functional Hubbard-Stratonovich transformation to analytically compute the mean density of states $\rho(E)$ in the large-$N$ limit for both Lévy and Lévy-RP ensembles. A key result is a closed representation of $\rho(E)$ through a transcendental equation for $f_{\mu}(E)$, together with a single-parameter scaling form controlled by $\xi = W^{-1} N^{(1-\gamma)/\mu}$ and an order parameter $\tilde{\rho}(0)=\rho(0) N^{(1-\gamma)/\mu}$ signaling the ergodic transition; these findings are validated by numerical diagonalization up to large $N$. The approach provides a fast, general methodology for heavy-tailed random Hamiltonians and clarifies finite-size scaling at the ergodic transition, with implications for non-ergodic extended states in disordered systems. The results offer a robust theoretical framework for analyzing spectral properties in systems with fat-tailed randomness and potential applications to MBL-related phenomena.

Abstract

By using the Efetov's super-symmetric formalism we computed analytically the mean spectral density $ρ(E)$ for the Lévy and the Lévy -Rosenzweig-Porter random matrices which off-diagonal elements are strongly non-Gaussian with power-law tails. This makes the standard Hubbard-Stratonovich transformation inapplicable to such problems. We used, instead, the functional Hubbard-Stratonovich transformation which allowed to solve the problem analytically for large sizes of matrices. We show that $ρ(E)$ depends crucially on the control parameter that drives the system through the transition between the ergodic and the fractal phases and it can be used as an order parameter.

Figures (4)

• Figure 1: DoS $\rho(E)$ at the ergodic transition for the Lévy-RP random matrices: (a) The diverging with $N$ band-width $B\propto N^{(1-\gamma)/\mu}$ in the ergodic phase, $\gamma=0.7$, the $N$-independent DoS at the ergodic transition, $\gamma=1$, and a convergent with increasing $N$ DoS in the non-ergodic extended phase $\gamma=1.3$. (b) Dependence on $N$ of the "order parameter" $\rho(0) N^{\frac{1-\gamma }{\mu}}$ plotted as a function of $\tau=\gamma_{c}-\gamma$. (c) Single-parameter scaling: all curves for $\rho(0)N^{\frac{1-\gamma }{\mu}}$ as functions of $\ln \xi = \ln(W^{-1}N^{(1-\gamma)/\mu})$ for different disorder strength $W$ and different matrix sizes $N$ collapse to a single curve which depends only on $\mu$. All plots are obtained from our analytical results, Eqs.(\ref{['f_rescaled']}),(\ref{['rho_rescaled']}).
• Figure 2: Phase diagram for the Lévy-RP model.
• Figure 3: Comparison of the analytical results of Eqs.(\ref{['f(mu, E) integral equation']}),(\ref{['rho(E) final']}) (shown by green lines) with the numerical diagonalization of the Lévy matrices for $N=5000$(10 different samples) and $\mu=1$(Cauchy), $\mu=3/2$, $\mu=2$(Gauss) (shown by histograms).
• Figure 4: Comparison of the analytical results of Eqs.(\ref{['f(mu, E) integral equation']}),(\ref{['rho(E) final']}) (shown by green lines) with the numerical diagonalization of the Lévy-RP matrices for $\gamma=1$, $W=1$, $N=5000$(10 samples) and $\mu=1$, $\mu=1.5$, $\mu=2$ (shown by histograms).