Circular Rosenzweig-Porter random matrix ensemble

TL;DR

The circular Rosenzweig-Porter (CRP) ensemble is introduced as a unitary analogue of the RP model to describe level statistics and eigenstate fractality across MBL-like transitions in periodically driven (Floquet) systems. The CRP is constructed as the outcome of a circular Dyson Brownian motion on a symmetric unitary matrix S(t) with eigenphases, initialized with uniform phases and evolved to t = \epsilon^2 / N^\gamma, yielding a numerically tractable model for Floquet MBL phenomenology. Numerical results show RP-like behavior in the middle of the spectrum, including a Wigner-Dyson to Poisson transition at \gamma_c = 2 (with \nu = 1), fractal eigenstate structure (IPR_2 scaling) and Breit-Wigner eigenstate amplitudes, indicating that CRP captures essential statistics of RP in a unitary setting. This unitary framework provides a stepping stone toward multifractal generalizations and potential applications to random quantum circuits and Floquet-era localization phenomena.

Abstract

The Rosenzweig-Porter random matrix ensemble serves as a qualitative phenomenological model for the level statistics and fractality of eigenstates across the many-body localization transition in static systems. We propose a unitary (circular) analogue of this ensemble, which similarly captures the phenomenology of many-body localization in periodically driven (Floquet) systems. We define this ensemble as the outcome of a Dyson Brownian motion process. We show numerical evidence that this ensemble shares some key statistical properties with the Rosenzweig-Porter ensemble for both the eigenvalues and the eigenstates.

Figures (3)

• Figure 1: The average ratio of consecutive level spacing spacings $\overline{r}$ as a function of $\gamma$ (upper panel) and $(\gamma - \gamma_c) \ln(N)^{1 / \nu}$ with $\gamma_c = 2$ and $\nu = 1$ (lower panel) for matrix dimensions $N=250$, $500$, and $1000$. Poissonian and Wigner-Dyson level statistics are characterized by respectively $\overline{r} \approx 0.386$ and $\overline{r} \approx 0.530$ (dashed lines).
• Figure 2: The average of the inverse participation ratio $\text{IPR}_2$ (upper panel) and the value scaled by $N$ (lower panel) as a function of $\gamma$ for matrix dimensions $N=250$, $500$, and $1000$.
• Figure 3: The average of $| \langle m | \psi_n \rangle |^2$ as a function of $\theta_n - \theta_m^{(0)}$ (markers), combined with the evaluation of Eq. \ref{['eq: BW-theta']} with $\rho = 1000 / (2 \pi)$ (red solid lines) for $N = 1000$ with $\gamma = 0.5$ (upper panel) and $\gamma = 1.5$ (lower panel). Each of the curves is normalized to unit area over the plotted range. Blue lines connecting the markers serve as a guide to the eye.