We study numerically the onset of chaos and thermalization in the Banks-Fischler-Shenker-Susskind (BFSS) matrix model with and without fermions, considering Lyapunov exponents, entanglement generation, and quasinormal ringing. We approximate the real-time dynamics in terms of the most general Gaussian density matrices with parameters which obey self-consistent equations of motion, thus extending the applicability of real-time simulations beyond the classical limit. Initial values of these Gaussian density matrices are optimized to be as close as possible to the thermal equilibrium state of the system. Thus attempting to bridge between the low-energy regime with a calculable holographic description and the classical regime at high energies, we find that quantum corrections to classical dynamics tend to decrease the Lyapunov exponents, which is essential for consistency with the Maldacena-Shenker-Stanford (MSS) bound at low temperatures. The entanglement entropy is found to exhibit an expected "scrambling" behavior - rapid initial growth followed by saturation. At least at high temperatures the entanglement saturation time appears to be governed by classical Lyapunov exponents. Decay of quasinormal modes is found to be characterized by the shortest time scale of all. We also find that while the bosonic matrix model becomes non-chaotic in the low-temperature regime, for the full BFSS model with fermions the leading Lyapunov exponent, entanglement saturation time, and decay rate of quasinormal modes all remain finite and non-zero down to the lowest temperatures.
Microcanonical Operator Spectrum, BPS Chaos and Free Probability Theory
Dipartimento di Fisica, Università di Milano - Bicocca（米兰比可卡大学物理系）；INFN, sezione di Milano-Bicocca（意大利国家核物理研究所米兰比可卡分部）；Department of Physics and Photon Science, Gwangju Institute of Science and Technology（光州科学技术院物理与光子科学系）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
Deep thermalization with and without quantum chaos
University of Chicago（芝加哥大学）；Massachusetts Institute of Technology（麻省理工学院）；Simons Institute for the Theory of Computing, UC Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；Stanford University（斯坦福大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学量子信息与计算机科学联合中心）
Characterizing complexity of many-body quantum dynamics by higher-order eigenstate thermalization
动力学的复杂性是量子多体混沌的核心，并表现出一个层次性的功能：高阶复杂性意味着更多的混沌动力学。热化过程中的遍历性是最低阶复杂性的表现，其由本征态热化假设（ETH）表示，该假设指出单个能量本征态是热的。在这里，我们提出了ETH的高阶推广，命名为$ k $（$ k=1,2,\dots $），以在单个能量本征态的水平上量化量子多体动力学的高阶复杂性，其中最低阶ETH（1-ETH）是常规ETH。作为高阶ETH的一个重要贡献，我们证明了具有$ k\geq 2 $的$ k $-ETH暗示了单个能量本征态的$ k $ th Renyi纠缠熵的普遍行为。特别地，纠缠熵的Page校正源自高阶ETH，而众所周知，体积定律可以由1-ETH解释。我们数值验证了2-ETH近似适用于不可积系统，但不适用于可积系统。为了进一步研究$ k $ -ETH背后的信息理论特征，我们引入了一个名为部分酉$ k $ -设计（PU $ k $ -设计）的概念，它是Haar随机酉到$ k $ th时刻的近似，其中部分意味着只有有限数量的可观测量。$ k $ -ETH是PU $ k $ -设计的特殊情况，用于具有随机时间采样的哈密顿动力学系综。此外，我们还讨论了高阶ETH和由非时间排序的置乱器量化的信息置乱之间的关系。我们的框架提供了一个统一的看法热化，纠缠熵，酉$ k $设计，导致更深层次的表征高阶量子复杂性。
