Semiclassical Approach to Spectral Statistics of Interacting Quantum Many-Body Systems

近年来，相互作用多体系统的研究取得了长足的进展。在理论方面对多体定域的进一步理解与最近的冷原子实验相结合，允许制造和控制相互作用的多体系统。更好地理解相互作用多体系统的动力学是凝聚态物理、核物理和量子光学等许多研究领域的一个中心任务，本文的目标是描述相互作用量子多体系统的谱统计（即谱关联）。理解单粒子系统的谱统计和经典动力学之间的联系是近几十年来量子混沌研究领域的一个热点。然而，对于多体系统来说，由于粒子的不可逆性和玻色子或费米子性质，新的特征出现了。对于光谱统计来说特别重要的是集体运动的问题，而不是单个粒子的非相干运动。更深入的理论理解是困难的，因为在量子混沌中发展的推理路线在考虑集体性时不能简单地从一个粒子转移到多个粒子。基于我们最近的工作，我们想研究这个具有挑战性的问题，从目前在理论和实验中受到极大关注的自旋链开始。我们的目标也是在第一步，将我们的研究扩展到更广泛的一类相互作用的多体系统。

In recent years, the study of interacting many-body systems has made considerable progress. An improved understanding of many-body localization on the theoretical sideis paired with recent cold-atom experiments allowing the fabrication and the control of interacting many-body systems. To comprehend ever better the dynamics of interacting many-body systems is a central task in many research fields including condensed matter physics, nuclear physics and quantum optics.The aim of this proposal is to characterize the spectral statistics (i.e. spectral correlations) of interacting quantum many-body systems. Understanding the connection between the spectral statistics and the classical dynamics for single-particle systems has been subject of intensive research during the last decades in the research field of quantum chaos. However, for a many-body system new features arise for example due to indistinguishability and the bosonic or fermionic character of the particles. Particularly important for the spectral statistics is also the issue of collective motion as opposed to incoherent motion of the individual particles. A deeper theoretical understanding is difficult as the lines of reasoning developed in quantum chaos cannot simply be carried over from one to many particles when considering collectivity. Building on our recent work we want to study this challenging problem, starting from spin chains which presently receive much interest in theory and experiment. We also aim at first steps to extend our study to a broader class of interacting many-body systems.