谱统计、流体力学与量子混沌
Spectral Statistics, Hydrodynamics and Quantum Chaos
AI总结：
本论文研究多体物理中热化与谱混沌两种混沌概念的关联，定义总返回概率（TRP）并证明其与谱形状因子（SFF）的联系，在流体力学系统和自旋玻璃中解析验证了二者的关联，为混沌研究提供新方向。
摘要：
多体物理（包括经典与量子领域）的核心问题之一，是不同混沌概念之间的关联。遍历性、混合性、算符增长、本征态热化假说以及谱混沌，均在不同语境下基于完全不同的对象定义，它们未必同时出现，但似乎仍是紧密相关现象的体现。本论文研究了两种混沌概念——热化与谱混沌——之间的关联。我们定义了一个名为总返回概率（Total Return Probability，TRP）的物理量，用于衡量系统在时间$T$后对初始状态的遗忘程度，并证明它与谱形状因子（Spectral Form Factor，SFF）密切相关，后者是一种源自量子系统能级谱的混沌度量。本工作的核心聚焦于流体力学系统——在这类系统中，局域性会阻碍电荷或能量快速扩散，从而限制热化进程。我们证明，能级的精细间距能够精准捕捉这些局域守恒荷的动力学行为。我们还研究了自旋玻璃（spin glasses）这一物质相，其中阻碍热化的因素并非局域性，而是近邻数量过多：改变一个区域需要改变邻近区域，进而需要改变邻近区域的周边区域，最终只有整个系统发生彻底重构，才能充分探索相空间。在自旋玻璃体系中，我们找到了热化与谱统计之间最清晰的解析关联。我们在大系统尺寸极限下解析计算了谱形状因子，并证明其等于总返回概率。最后，在结论部分，我们讨论了SFF与TRP未来的若干研究方向。
英文摘要：
One of the central problems in many-body physics, both classical and quantum, is the relations between different notions of chaos. Ergodicity, mixing, operator growth, the eigenstate thermalization hypothesis, and spectral chaos are defined in terms of completely different objects in different contexts, don't necessarily co-occur, but still seem to be manifestations of closely related phenomena. In this dissertation, we study the relation between two notions of chaos: thermalization and spectral chaos. We define a quantity called the Total Return Probability (TRP) which measures how a system forgets its initial state after time $T$, and show that it is closely connected to the Spectral Form Factor (SFF), a measure of chaos deriving from the energy level spectrum of a quantum system. The main thrust of this work concerns hydrodynamic systems -- systems where locality prevents charge or energy from spreading quickly, this putting a throttle on thermalization. We show that the detailed spacings of energy levels closely capture the dynamics of these locally conserved charges. We also study spin glasses, a phase of matter where the obstacle to thermalization comes not from locality but from the presence of too many neighbors. Changing one region requires changing nearby regions which requires changing nearby-to-nearby regions, until only catastrophic realignments of the whole system can fully explore phase space. In spin glasses we find our clearest analytic link between thermalization and spectral statistics. We analytically calculate the spectral form factor in the limit of large system size and show it is equal to the TRP. Finally, in the conclusion, we discuss some ideas for the future of both the SFF and the TRP.
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