Nonequilibrium dynamics in 2D clusters from the perspective of quantum typicality and eigenstate thermalization

理解失去平衡的多体系统物理学的基本方面既是非常理想的，也是出了名的困难。特别是，解释孤立量子系统中统计行为的发生仍然是对理论的挑战，包括从真正的微观原理推导出无处不在的现象，如指数松弛、扩散、平衡和热化。最近在这方面取得的实质性进展也与量子典型性的新兴概念和著名的本征态热化假说(ETH)有关。量子典型性本质上是指从高维Hilbert空间中随机抽取的单个纯态的期望值可以准确地给出系综平均值，而量子典型性的对角部分甚至在单个能量本征态的水平上也具有这种典型性。在这个项目中，我们在前人工作的基础上，例如，我们在本单位第一个成立时期的工作基础上，计划从量子典型性和ETH的角度来研究二维(2D)团簇中的非平衡动力学。在我们的研究中，我们将严重依赖动态量子典型性(DQT)和最先进的(超级)计算来处理这些2D集群，范围从矩形几何中相互作用的自旋的有限系统，如两条腿或多条腿的梯子，到正方形晶格。此外，我们将结合DQT和投影算子技术以及数值链簇展开(NCLE)来补充我们的分析。尤其是DQT和NLCE的后一种结合，对于揭示扩展的2D模型的物理学有很大的潜力，不管有没有无序。这样，我们也许能够在更高的维度上对多体局部化问题做出贡献。虽然我们的项目中有大量的研究致力于线性响应理论框架内的运输量，但另一个同样重要的问题是特定初始条件对随后松弛过程的作用。在此背景下，我们将通过进行超越随机矩阵型理想化模型的几个案例研究来仔细研究我们目前的建议，即ETH ansatz的(非对角线部分)意味着远离和接近平衡的动力学之间的“普遍”关系。在同样的背景下，我们对非输运类型的可观测数据的扩展也允许我们探索基于典型性的对整个松弛过程的预测，本质上与生存概率有关，在多大程度上适用于一般的非平衡情况。

Understanding fundamental aspects of the physics of many-body systems out of equilibrium is both, highly desired and notoriously difficult. In particular, explaining the occurrence of statistical behavior in isolated quantum systems continues to be a challenge to theory, including the derivation of omnipresent phenomena such as exponential relaxation, diffusion, equilibration, and thermalization from truly microscopic principles. Recent and substantial progress in this context is also related to the emergent concept of quantum typicality and the celebrated eigenstate thermalization hypothesis (ETH). While quantum typicality means in essence that an ensemble average is accurately given by an expectation value of just a single pure state drawn at random from a high-dimensional Hilbert space, the (diagonal part of the) ETH ansatz assumes such type of typicality even on the level of individual energy eigenstates.In this project, we build on our previous works, e.g., on our works in the first fundingperiod of this research unit, and plan to study nonequilibrium dynamics in two-dimensional (2D) clusters from the perspective of quantum typicality and ETH. In our study, we will heavily rely on the use of dynamical quantum typicality (DQT) and state-of-the-art (super) computing to tackle these 2D clusters, ranging from finite systems of interacting spins in a rectangular geometry, like two-leg or multi-leg ladders, to square lattices. Moreover, we will complement our analysis by a combination of DQT with projection operator techniques and numerical linked-cluster expansions (NCLE). Especially the latter combination of DQT with NLCE has the promising potential to shed light onto the physics of extended 2D models, with or without disorder. In this way, we might be able to contribute to the question of many-body localization in higher dimensions.While a significant body of research in our project is devoted to transport quantities withinthe framework of linear response theory, another equally important question constitutes the role of the specific initial condition for the subsequent relaxation process. In this context, we will scrutinize our current proposal that the (off-diagonal part of the) ETH ansatz implies a "universal" relation between the dynamics far away from and close to equilibrium, by performing several cases studies beyond idealized models of random-matrix type. In the same context, our extension to observables of non-transport type also allows us to explore in how far typicality-based predictions for the entire relaxation process, essentially related to survival probabilities, apply to generic nonequilibrium situations.