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. 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. Using the distance measures and statistical methods, we define and characterize the onset of the free time in the ergodic phase. The talk is based on arXiv: 2506.04520.