Multibody quantum chaos in SYK and SYK-like models using semi-classical methods

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Recently the SYK and similar models have garnered huge interest due to their unique properties. They are analytically solvable in the limit of a large amount of Majorana fermion fields, represent a possible holographic dual to 2-dimensional gravity models and necessarily fulfil the upper bound of the quantum Lyapunov exponent. This latter property makes these models ideal for studying multibody quantum chaos. The measure of quantum chaos used for SYK models is derived using out-of-time-ordered correlators (OTOCs), which contrasts with the historic analysis of chaotic quantum systems.Analysing the properties of quantum systems whose underlying classical dynamics is chaotic has historically been carried out using the semi-classical limit . This has been an extremely fruitful approach and has led to e.g. new examples of periodic orbit theory (such as Sieber-Richter pairs), Gutzwiller's trace formula and examples of the universality of chaotic energy spectrums. The majority of this work was concerned with single-body systems, but lately there has been results for multibody bosonic systems. However, currently there has not been a thorough attempt at analysing multibody fermionic systems using semi-classical techniques. For SYK models this would represent a change in approach, i.e. we are now asking questions about the classical dynamics of systems who are known to be chaotic in a quantum sense. The semi-classics of fermions is currently not well understood, with the main issues arising because of the anti-commuting nature of the particles in question. This property has no obvious classical analogue and any semi-classical work concerning fermions (whether chaotic/SYK-like or not) is highly likely to shed light upon the question of how to handle anti-commuting variables appearing in classical equations of motion. For general chaotic fermionic systems we would expect to derive new examples of the universality of the energy spectrum. It has already been shown, using random matrix theory (RMT), that we should expect to see repulsion of energy levels, as in the single particle and multi-particle bosonic cases. Deriving a trace formula for chaotic fermionic systems will provide us with a concrete connection to physical systems, as oppose to the more abstract RMT methodology. For SYK models specifically this work could also elucidate a connection between quantum chaos as defined through OTOCs and quantum chaos as defined by underlying classical dynamics.