The Wishart--Rosenzweig--Porter random matrix ensemble

TL;DR

We introduce the Wishart--Rosenzweig--Porter ensemble $\mathbf{H}=\mathbf{A}+\nu\mathbf{B}$ with $\mathbf{B}=M^{-\gamma}\mathbf{W}\mathbf{W}^T$ and analyze its spectrum across three phases: delocalized ($\gamma<1$), fractal intermediate ($1<\gamma<3/2$), and localized ($\gamma>3/2$). Using perturbation theory, the cavity method, the replica method, and Dyson Brownian motion, we show that the intermediate fractal phase exhibits a universal level compressibility crossover $\chi(y)=\frac{1}{\pi y}\bigl[2 y \arctan(y) - \ln(1+y^2)\bigr]$, identical to Gaussian RP, despite correlations in the Wishart block. The universality persists across distributions of $A$ and for correlated $B$, as supported by numerics. These results reinforce a robust, scale-dependent universality of spectral correlations in non-ergodic extended regimes and suggest broader implications for covariance denoising and diffusion-based generative modeling.

Abstract

In recent years the Rosenzweig--Porter (RP) ensemble, obtained by adding a diagonal matrix with independent and identically distributed elements to a Gaussian random matrix, has been widely used as a minimal model for the emergence of fractal eigenstates in complex many-body systems. A key open question concerns the robustness of its phase diagram when the assumption of independent and uncorrelated entries is relaxed -- an assumption that simplifies its analysis, but is generally violated in realistic quantum systems. In this work, we take a first step in this direction by considering a deformed Wishart (rather than Gaussian) random matrix, which we dub the ``Wishart--RP'' ensemble. Using perturbation theory, as well as the cavity and replica methods and the Dyson Brownian motion approach, we characterize its phase diagram and localization properties. Remarkably, we show that the level compressibility, which quantifies spectral correlations in the fractal phase, coincides with that of the Gaussian RP model, thereby extending the universality conjectured in [SciPost Phys. 14, 110 (2023)] beyond the fully uncorrelated setting. We confirm our results with numerical tests.

Figures (7)

• Figure 1: The phase diagram of the Wishart--Rosenzweig--Porter (WRP) model (see Sec. \ref{['sec:summary']} for its description).
• Figure 2: Illustration of the different scales in the spectrum in the intermediate phase of the WRP model ($1<\gamma < 3/2$). The blue window illustrates the concept of mini-bands in the spectrum, extending over the scale of the Thouless energy $E_T$.
• Figure 3: Level compressibility of the WRP ensemble. The numerical data obtained via exact diagonalization (green symbols) is compared to analytical predictions in the distinct regimes of the model, as detailed below. First, at small energy scales (comparable to the mean level spacing $\Delta = 1 / (N \rho)$), the level compressibility decreases in accordance with the universal behavior $\chi_{\mathrm{GOE}}$ of Wigner--Dyson statistics (see \ref{['chi_GOE_final']}). In this regime, the spectrum is said to be rigid because the variance of the number of eigenvalues in an interval is small due to level repulsion (and thus, so is the level compressibility). Conversely, for intervals of $\mathcal{O}(1)$, the level compressibility matches that of i.i.d. random variables \ref{['eq:chiPoisson']}, meaning that there are no correlations between energy levels at this scale. Finally, for intervals close to the Thouless energy $E_T$, the numerical curve approaches the (super)universal crossover function $\chi_T$ which connects the two regimes, see \ref{['eq:comp_universal']}. Note that the replica method provides an analytical prediction (yellow symbols, see Sec. \ref{['sec:replica']}) that is valid for all regimes, except the one at $E\lesssim \Delta$, where it breaks down. For this plot, we used $\gamma=1.25$, $c\approx 0.998$, $\nu= 1$, $N =2000$, and $p_a$ is the standard Cauchy distribution.
• Figure 4: Numerical data for the level compressibility of the WRP model at the Thouless energy scale, obtained by exact diagonalization of large random matrices, for (a) different values of $N$ with $c = 0.98$ fixed, and (b) different values of $c$ with $N= 20000$ fixed. We chose $\gamma=1.25$ and $\nu= 1$; moreover, we took $p_a$ to be uniform between $-1$ and $1$, which allowed us to average the level compressibility over multiple non-overlapping intervals (which are statistically equivalent since the density of states is constant). The dashed line indicates the universal scaling form $\chi_T$ given in \ref{['eq:comp_universal']}.
• Figure 5: (a) Multifractal spectrum $f(\alpha)$, Eq. \ref{['eq:falpha']}, and (b) fractal dimensions $D_q$, Eqs. \ref{['eq:Dfirst']} and \ref{['eq:Dloc']}, for several values of $\gamma$ spanning the three regimes.