Chaos and ergodicity are the cornerstones of statistical physics and thermodynamics. While classically even small systems like a particle in a two-dimensional cavity, can exhibit chaotic behavior and thereby relax to a microcanonical ensemble, quantum systems formally can not. Recent theoretical breakthroughs and, in particular, the eigenstate thermalization hypothesis (ETH) however indicate that quantum systems can also thermalize. In fact ETH provided us with a framework connecting microscopic models and macroscopic phenomena, based on the notion of highly entangled quantum states. Such thermalization was beautifully demonstrated experimentally by A. Kaufman et. al. who studied relaxation dynamics of a small lattice system of interacting bosonic particles. By directly measuring the entanglement entropy of subsystems, as well as other observables, they showed that after the initial transient time the system locally relaxes to a thermal ensemble while globally maintaining a zero-entropy pure state.
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。
