Microscopic Foundations of the Eigenstate Thermalization Hypothesis

对经典或量子力学多粒子系统建模的一个基本假设是，它们在足够长的等待时间后返回到平衡状态。然后，这种平衡态可以被描述为只有温度作为参数的吉布斯状态(在最简单的情况下)。所有多粒子系统性质的理论计算，如输运性质、热力学性质等，都建立在这个热化假设的基础上。这一假设的教科书推导是通过与大环境的耦合进行的。在过去的十年里，对超冷原子气体的实验成为可能，因为超冷原子气体与环境隔离得非常好，所以现在人们还需要解决封闭量子系统中的热化问题。对于不可积的一般量子多粒子系统(即它们只有有限数量的守恒量)，人们提出了所谓的本征态热化假设(ETH)来解释这种封闭的不可积系统的热化。关于这个主题的大多数出版物都是以数字为基础的，也就是说，他们为某些模型哈密尔顿建立了ETH。但值得一提的是，还有一些悬而未决的问题，因此即使是数字情况也不完全明朗。在这个项目中，我们的目标是寻求一种互补的方法，这至少是部分分析的。起始点是J.M.Deutsch[Phys]发表的一本较老的出版物。Rev.A43,2046(1991)]，他可以解析地证明，如果一个人使用随机矩阵模型，那么ETH是满足的。由于现实的微观哈密顿量不是随机矩阵，德伊奇的工作在目前的文献中扮演了一个不那么突出的角色。这个项目旨在通过使用一系列无限小的么正变换(Wegner流动方程)将现实的微观哈密顿量映射到随机矩阵哈密顿量来弥合这一差距。通过这种方式，J.M.Deutsch的分析论证路线可以拉回到最初的现实微观哈密顿。此外，对于哈密顿量和可观测性，特别是对于在空间和/或时间上非局域的关联函数，人们可以了解到一些关于ETH成立的条件。理想情况下，我们想要追求的解析方法可以作为各种数值方法的统一支架，从而有助于更深入地理解描述量子力学多粒子系统的热化假设。

A fundamental assumption for the modeling of classical or quantum mechanical many-particle systems is that they return to an equilibrium state after a sufficiently long waiting time. This equilibrium state can then be described as a Gibbs state with only the temperature as a parameter (in the simplest case). All theoretical calculations of properties of many-particle systems like transport properties, thermodynamic properties, etc., are built on this thermalization assumption. The textbook derivation of this assumption proceeds via the coupling to a large environment. In the past decade experiments in ultracold atomic gases, which are extremely well isolated from their environment, became possible so that now one also needs to address the question of thermalization in closed quantum systems. For generic quantum many-particle systems, which are non-integrable (meaning they have only a finite number of conserved quantities), the so called eigenstate thermalization hypothesis (ETH) has been put forward as an explanation for thermalization of such closed non-integrable systems. Most publications regarding this topic proceed numerically, that is they establish ETH for certain model Hamiltonians. However, it should be mentioned that there are unresolved questions, so even the numerical situation is not entirely clear. In this project we aim to pursue a complementary approach, which is at least partially analytic. The starting point is an older publication by J. M. Deutsch [Phys. Rev. A 43, 2046 (1991)], who could show analytically that ETH is fulfilled if one works with a random matrix model. Deutsch's work plays a somehow less prominent role in the current literature since realistic microscopic Hamiltonians are not random matrices. This project aims at closing this gap by mapping a realistic microscopic Hamiltonian to a random matrix Hamiltonian using a sequence of infinitesimal unitary transformations (Wegner flow equations). In this way the analytical line of argument by J. M. Deutsch can be pulled back to the original realistic microscopic Hamiltonian. Additionally, one can learn something about the conditions under which ETH holds, both for the Hamiltonian and the observables, especially also for correlations functions which are nonlocal in space and/or time. Ideally, the analytic approach that we want to pursue could serve as a unifying bracket for the various numerical approaches and thereby be a contribution to a deeper understanding of the thermalization assumption for the description of quantum mechanical many-particle systems.