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

最近，SYK和类似的型号由于其独特的性能而引起了巨大的兴趣。它们在大量Majorana费米子场的极限下是解析可解的，代表了一种可能的全息对偶到二维引力模型，并且必然满足量子Lyapunov指数的上界。后一种性质使这些模型成为研究多体量子混沌的理想模型。利用超时有序相关器(OTOCs)推导了用于SYK模型的量子混沌度量，与混沌量子系统的历史分析进行了对比，历史上使用半经典极限分析了其基本经典动力学为混沌的量子系统的性质。这是一种非常卓有成效的方法，并产生了周期轨道理论的新例子(如Sieber-Richter对)、古兹威勒迹公式和混沌能谱普适性的例子。这项工作的大部分是关于单体系统的，但最近也有了关于多体玻色子系统的结果。然而，目前还没有人尝试用半经典方法来分析多体费米子系统。对于SYK模型来说，这代表着方法的改变，也就是说，我们现在询问的是关于已知在量子意义上是混沌的系统的经典动力学问题。费米子的半经典目前还没有被很好地理解，主要的问题是由于所讨论的粒子的反通勤性质而引起的。这一性质没有明显的经典相似之处，任何关于费米子的半经典工作(无论是否类似混沌/类SYK)都很有可能揭示如何处理出现在经典运动方程中的反对易变量的问题。对于一般的混沌费米子系统，我们期望得到能谱普适性的新例子。用随机矩阵理论(RMT)已经证明，我们应该看到能级的排斥，就像在单粒子和多粒子玻色子的情况下一样。推导出混沌费米子系统的迹公式将为我们提供与物理系统的具体联系，而不是更抽象的RMT方法。具体地说，对于SYK模型，这项工作还可以阐明通过OTOC定义的量子混沌和由潜在经典动力学定义的量子混沌之间的联系。

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.