Chaos and Scrambling in many-body quantum systems

一个长期存在的问题是理解量子多体系统中的混沌。特别是，虽然混沌在经典动力学和单粒子量子力学中早已被理解，但在多体量子系统中理解混沌的含义仍然极具挑战性。然而，量子混沌的问题对于理解多体量子动力学至关重要，尤其是因为它与量子力学系统如何热化有关。在过去的几年里，通过信息置乱、失时有序子的衰减和有效的算子扩展的特性，提供了多体量子系统中混沌的新特性。该项目的目的是研究多体量子系统中混沌和加扰的新特征，使用全息量子理论和量子信息理论的技术。特别是，在过去的几年里，多体量子系统中的混乱和流体力学之间出现了令人惊讶的联系，包括能量扩散和混沌之间的惊人联系。基于这些联系，[1]提出了一种有效的流体动力学理论，用于最大混沌系统中的置乱，其中算子的置乱是由于与流体动力学自由度的相互作用而产生的。该理论的一个成果是预测了最大混沌系统中的一种新现象，称为极点跳跃。这是指出现在时间外有序相关函数中的李雅普诺夫指数和蝴蝶速度与系统的流体动力学模式（集体激励）的色散关系之间的精确连接。检验这种现象的一个重要工具是全息量子场论。这类场论在反德西特空间中对黑洞的动力学有双重的数学描述，利用它可以计算这些量子场论的超时有序关联函数和流体力学模。本项目的初步目标是研究上述的跳极现象，在量子系统的背景下，这是全息对偶的旋转迈尔斯-佩里-AdS黑洞。未来的工作可能会探索多体量子系统中混沌的其他方面。例如，这可能包括研究多体量子系统中的纠缠动力学或研究其他玩具模型（例如量子电路）中的加扰。

A long standing issue has been to understand chaos in quantum many body systems. In particular, whilst chaos has long been understood in classical dynamics and in single-particle quantum mechanics, it has remained extremely challenging to understand what is mean by chaos in a a many-body quantum system. Nevertheless, the question of quantum chaos is of central importance in understanding many-body quantum dynamics, not least due its connection to how quantum mechanical systems thermalise. Over the last several years, a new characterisation of chaos in many-body quantum systems has been provided through a characterisation of information scrambling, the decay of out of time ordered correlators, and efficient operator spreading. The aim of this project will be to study new signatures of chaos and scrambling in many-body quantum systems, using techniques from holographic quantum theories and and quantum information theory. In particular, over the last few years there have emerged surprising connections between scrambling in many-body quantum systems and hydrodynamics, including surprising connections between energy diffusion and chaos. Based on these connections, [1] proposed an effective hydrodynamic theory for scrambling in maximally chaotic systems, in which the scrambling of operators arises due to interactions with hydrodynamic degrees of freedom. One output of this theory was a prediction of a new phenomenon in maximally chaotic systems known as pole-skipping. This refers to a precise connection between the Lyapunov exponent and butterfly velocity that appear in out-of-time ordered correlation functions and the dispersion relations of hydrodynamic modes (collective excitations) of the system. An important tool for testing this phenomenon are holographic quantum field theories. Such field theories have a dual mathematical description in terms of the dynamics of black holes in anti-de Sitter space, using which one can compute both out-of-time ordered correlation functions and hydrodynamic modes of these quantum field theories.The initial aim of this project will be to study the pole-skipping phenomenon mentioned above, in the context of quantum systems which are holographically dual to the rotating Myers-Perry-AdS black hole. Future work is likely to explore other aspects of chaos in many-body quantum systems. For example, this could include studying entanglement dynamics in many-body quantum systems or studying scrambling in other toy models e.g. quantum circuits.