A free probability approach to quantum chaos in random matrix ensembles

In free probability theory, quantum chaos is marked by "free independence" between observables at early and late times, causing certain statistical measures (cumulants) to vanish. Motivated by this, we study the statistics of a time-evolved operator in the Rosenzweig-Porter (RP) random matrix ensembles. The RP model displays a complex phase structure, including ergodic, fractal, and localized regimes. Analyzing operator statistics for different spin operators across these regimes reveals close alignment with free probability predictions in the ergodic phase, contrasted by persistent deviations in the fractal and localized phases even at late times. Notably, the fractal phase exhibits partial characteristics of freeness while retaining memories of the initial spectrum. Using the distance measures and statistical methods such as the $\chi^2$ function, Kullback-Leibler divergence, and Kolmogorov-Smirnov hypothesis testing, we define and characterize the onset of the free time in the ergodic phase. Remarkably, our findings demonstrate consistent results across different approaches.