Onset of Random Matrix Statistics in Scrambling Systems

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The fine grained energy spectrum of quantum chaotic systems, which are widely believed to be characterized by random matrix statistics. A basic scale in these systems is the energy range over which this behavior persists. We defined the corresponding time scale by the time at which the linearly growing ramp region in the spectral form factor begins. We dubbed this ramp time. It is also referred to as the ergodic or Thouless time in the condensed matter physics community. The purpose of my talk is to understand this scale in many-body quantum systems that display strong chaos (such as SYK and spin chain), sometimes referred to as scrambling systems. Using numerical results and analytic estimates for random quantum circuits, I will provide summary of results on scaling of ramp time with system size in the presence/absence of conservation laws.