Microscopic Foundations of the Eigenstate Thermalization Hypothesis

本征态热化假说的微观基础

• 批准号: 255134628
• 负责人: Professor Dr. Stefan Kehrein
• 金额: --
• 依托单位: Institut für Theoretische Physik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2017-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/255134628?language=en
• 关键词: Microscopic; Foundations; Eigenstate; Thermalization; Hypothesis

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.

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。这种平衡状态可以被描述为吉布斯状态，只有温度作为参数（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，都是建立在热化假设上的教科书中对这一假设的推导是通过耦合到大环境来进行的。在过去的十年中，在与环境隔离得非常好的超冷原子气体中进行实验成为可能，因此现在还需要解决封闭量子系统中的热化问题。对于一般的量子多粒子系统，这是不可积的（这意味着他们只有有限数量的守恒量），所谓的本征态热化假设（ETH）已被提出来解释这种封闭的不可积系统的热化。大多数关于这个主题的出版物都是数值化的，也就是说，它们为某些模型哈密顿建立了ETH。然而，应该指出，还有一些问题没有解决，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。起点是J. M.多伊奇[Phys. Rev. A 43，2046（1991）]，他可以分析地表明，如果使用随机矩阵模型，ETH是满足的。多伊奇的工作起着某种不太突出的作用，在目前的文学，因为现实的微观哈密顿不是随机矩阵。这个项目的目的是缩小这一差距，映射一个现实的微观哈密顿随机矩阵哈密顿使用一系列的无穷小酉变换（韦格纳流方程）。这样，J.M.多伊奇可以被拉回到原来的现实微观哈密顿。此外，我们还可以了解ETH成立的条件，包括哈密顿量和可观测量，特别是在空间和/或时间上非局部的相关函数。理想情况下，我们想要追求的分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量子力学多粒子系统描述的热化假设。