Microscopic Foundations of the Eigenstate Thermalization Hypothesis

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会回到平衡状态。在最简单的情况下，这个平衡态可以只用温度作为参数来描述为吉布斯态。所有关于多粒子系统性质的理论计算，如输运性质、热力学性质等，都是建立在这种热化假设之上的。课本上对这个假设的推导是通过对一个大环境的耦合进行的。在过去的十年中，对超冷原子气体的实验成为可能，这些气体与环境非常隔绝，因此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（即它们只有有限个守恒量），提出了本征态热化假设（ETH）来解释这种封闭不可积系统的热化现象。关于这一主题的大多数出版物都是数值化的，即他们为某些模型哈密顿量建立ETH。然而，应该提到的是，仍有一些问题尚未解决，因此，即使是数字情况也不完全清楚。在这个项目中，我们的目标是追求一种互补的方法，这至少是部分分析。本文的出发点是j.m. Deutsch[物理学家]的一篇较早的出版物。Rev. A 43,2046(1991)]，他可以分析地表明，如果一个人使用随机矩阵模型，ETH是满足的。多伊奇的工作在当前的文献中扮演着不那么重要的角色，因为现实的微观哈密顿量不是随机矩阵。该项目旨在通过使用一系列无穷小的幺正变换（韦格纳流方程）将现实的微观哈密顿量映射到随机矩阵哈密顿量来缩小这一差距。这样，多伊奇的分析论证路线就可以被拉回原来现实的微观哈密顿量。此外，人们可以了解ETH的条件，无论是对于哈密顿量还是可观测量，特别是对于空间和/或时间上非局部的相关函数。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.