Abstract
We consider Random Matrix Theories with non-Gaussian potentials that have a rich phase structure in the large $N$ limit. We calculate the Spectral Form Factor (SFF) in such models and present them as interesting examples of dynamical models that display multi-criticality at short time-scales and universality at large time scales. The models with quartic and sextic potentials are explicitly worked out. The disconnected part of the Spectral Form Factor (SFF) shows a change in its decay behavior exactly at the critical points of each model. The dip-time of the SFF is estimated in each of these models. The late time behavior of all polynomial potential matrix models is shown to display a certain universality. This is related to the universality in the short distance correlations of the mean-level densities. We speculate on the implications of such universality for chaotic quantum systems including the SYK model.
Highlights
Quantum chaos in many-body systems has attracted the attention of physicists across various disciplines for many decades
(3) The dip time estimate changes with the change in the falloff behavior at criticality, providing an upper bound on the ramp time
We have studied the Spectral Form Factor (SFF) in nonGaussian random matrix models
Summary
Quantum chaos in many-body systems has attracted the attention of physicists across various disciplines for many decades now. An important diagnosis for chaos in classical systems is the sensitive dependence on initial conditions, characterized by a positive Lyapunov exponent An analog of this definition for a few-body quantum system was outlined in Ref. The SFF (when averaged over Gaussian random matrices) has a very particular behavior at large N characterized by an initial decay, followed by a linear rise and saturation This indicates how the SFF perceives the nearest-neighbor energy spectrum. We will try to understand how such deformations affect the chaotic structure of the energy eigenvalues This shall be reflected in any changes in the onset time for the ramp in the SFF. V, we talk about the implications of non-Gaussian matrix models on chaotic systems
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