Free Probability approach to spectral and operator statistics in Rosenzweig-Porter random matrix ensembles

TL;DR

The paper employs free probability to study spectral and operator statistics in the Rosenzweig-Porter random matrix ensemble, revealing a rich phase structure with ergodic, fractal, and localized regimes. Using subordination formulae and a perturbative resolvent scheme, it derives semi-analytic DOS corrections and examines higher-point correlators, linking their behavior to asymptotic freeness. In the ergodic phase, operator statistics align with free-probability predictions, while fractal and localized phases exhibit memory effects and deviations, with a well-defined free time marking the onset of freeness. The work also investigates the role of eigenvectors versus eigenvalues via decorrelated ensembles and KS/χ^2/KL metrics, highlighting how delocalized states drive freeness and how non-localized states preserve memory in late-time dynamics. Overall, the results establish a robust framework connecting spectral statistics, operator growth, and information scrambling in chaotic quantum systems through free probability.

Abstract

Utilizing the framework of free probability, we analyze the spectral and operator statistics of the Rosenzweig-Porter random matrix ensembles, which exhibit a rich phase structure encompassing ergodic, fractal, and localized regimes. Leveraging subordination formulae, we develop a perturbative scheme that yields semi-analytic expressions for the density of states up to second order in system size, in good agreement with numerical results. We compute higher-point correlation functions in the ergodic regime using both numerical and suitable analytic approximations. Our analysis of operator statistics for various spin operators across these regimes reveals close agreement with free probability predictions in the ergodic phase, in contrast to persistent deviations observed in the fractal and localized phases, even at late times. Notably, the fractal phase exhibits partial freeness while retaining memory of the initial spectrum, highlighting the importance of non-localized eigenstates and associated with the late-time dynamics of cumulative out-of-time-ordered-correlators (OTOCs). Employing distance measures and statistical tools such as the $χ^2$ statistic, Kullback-Leibler divergence, and Kolmogorov-Smirnov hypothesis testing, we define a characteristic time scale-the free time-that marks the onset of the validity of free probability predictions for operator spectral statistics in the ergodic phase. Remarkably, our findings demonstrate consistency across these different approaches.

Figures (26)

• Figure 1: The operator eigenvalue distributions predicted by the free probability for the addition of two free spin $s =1/2$ (left), $s = 1$ (middle), and $s = 3/2$ (right) operators. The spin $1/2$ and the spin $1$ distribution is given by the arcsine distribution \ref{['arcsinedist']} and distribution \ref{['eq: sumdis_spin1']}, respectively. In contrast, the distribution for spin $3/2$ can be evaluated using the subordination method.
• Figure 2: (a) The level spacing distribution $p(\xi)$ in the ergodic ($\gamma = 0)$, fractal ($\gamma = 1.5$), and localized ($\gamma = 5$) regimes. The solid blue and red lines denote the GOE and the Poisson distribution. The system size is $N = 2^{10}$, with a $10^4$ ensemble of Hamiltonians taken. For the spacing, only the bulk of the eigenvalue spectrum ($20\%$ around each side of the mid-spectrum) is considered. (b) The $r$-ratio of the RP model across different phases for different sizes of the matrices $N = 2^8 \,(5 \times 10^4),\, 2^{10} \,(10^4)$ and $2^{12} \,(10^2)$. The ensemble averages are shown in parentheses. The inset shows the data collapse, which clearly marks the fractal to the localized transition at $\gamma_c = 2$.
• Figure 3: Schematic representation of the DOS in the RP model across its three phases: the ergodic phase (left), characterized by a Wigner semicircle distribution; the fractal phase (center), showing an intermediate form between semicircular and Gaussian; and the localized phase (right), where the DOS is well approximated by a Gaussian distribution.
• Figure 4: DOS of the RP model: comparison of approximate analytical expression with numerical results. The blue and black-dashed curves respectively represent the analytical DOS with first and second-order corrections. The red curve represents the Gaussian eigenvalue distribution of the matrix $A$. Here we have set $s=1/2,~\gamma=1.1, L=10$, and the histogram shows the numerical DOS averaged over 5000 independent realisations of the RP model Hamiltonian.
• Figure 5: Two-point function $F_2^{(\gamma)}(t)$ (left) and four-point OTOC $F_4^{(\gamma)}(t)$ (right), shown in the ergodic phase (top row) and in the non-ergodic phases (bottom row). Parameters are set to $i=j=1$, $L=8$, and results are averaged over 100 realizations of the Hamiltonian.