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First, we focus on chaotic eigenstates and establish the so-called subsystem ETH and the Page curve as consequences of our construction. We also improve known calculations for thermal reduced density matrices and comment on an inherently free probabilistic aspect of the replica approach to entanglement entropy previously noticed in a calculation for the Page curve of an evaporating black hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a sufficient mechanism for thermalization, in general. In particular, we show that reduced density matrices relax to their equilibrium form and that systems obey the Page curve at late times. We also demonstrate that the different phases of entanglement growth are encoded in higher correlations of the EB. Lastly, we examine the chaotic structure of eigenstates and operators together and reveal previously overlooked correlations between them. 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引用次数: 0
摘要
特征态热化假说(ETH)是在一般孤立量子系统中出现统计力学的主要猜想,它是根据算子的矩阵元素提出的。被称为 "遍历双分区"(EB)的类似理论描述了纠缠和局域性,并以特征态的分量来表述。在本文中,我们对 EB 进行了重大概括,并将其与 ETH 统一起来,将 EB 扩展到研究更高的相关性和非平衡系统。我们的主要成果是一种图解形式主义,它基于最近发现的 ETH 与自由概率论之间的联系,计算特征状态与算子之间的任意相关性。我们把图中的连接成分称为广义自由积。我们从几个方面应用我们的形式主义。首先,我们专注于混沌特征状态,并建立了所谓的子系统 ETH 和佩奇曲线,作为我们构造的结果。我们还改进了已知的热还原密度矩阵计算,并对之前在蒸发黑洞佩奇曲线计算中注意到的纠缠熵复制方法的内在自由概率方面进行了评论。接下来,我们转向混沌量子动力学,并证明 ETH 是一般热化的充分机制。特别是,我们证明了还原密度矩阵会松弛到其平衡形式,并且系统在晚期服从佩奇曲线。我们还证明,纠缠增长的不同阶段是由 EB 的较高相关性编码的。最后,我们一起研究了特征状态和算子的混沌结构,并揭示了它们之间以前被忽视的相关性。最重要的是,这些相关性编码了蝴蝶速度,这是相互作用量子系统的一个众所周知的动力学特性。
Generalized free cumulants for quantum chaotic systems
The eigenstate thermalization hypothesis (ETH) is the leading conjecture for the emergence of statistical mechanics in generic isolated quantum systems and is formulated in terms of the matrix elements of operators. An analog known as the ergodic bipartition (EB) describes entanglement and locality and is formulated in terms of the components of eigenstates. In this paper, we significantly generalize the EB and unify it with the ETH, extending the EB to study higher correlations and systems out of equilibrium. Our main result is a diagrammatic formalism that computes arbitrary correlations between eigenstates and operators based on a recently uncovered connection between the ETH and free probability theory. We refer to the connected components of our diagrams as generalized free cumulants. We apply our formalism in several ways. First, we focus on chaotic eigenstates and establish the so-called subsystem ETH and the Page curve as consequences of our construction. We also improve known calculations for thermal reduced density matrices and comment on an inherently free probabilistic aspect of the replica approach to entanglement entropy previously noticed in a calculation for the Page curve of an evaporating black hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a sufficient mechanism for thermalization, in general. In particular, we show that reduced density matrices relax to their equilibrium form and that systems obey the Page curve at late times. We also demonstrate that the different phases of entanglement growth are encoded in higher correlations of the EB. Lastly, we examine the chaotic structure of eigenstates and operators together and reveal previously overlooked correlations between them. Crucially, these correlations encode butterfly velocities, a well-known dynamical property of interacting quantum systems.
期刊介绍:
The aim of the Journal of High Energy Physics (JHEP) is to ensure fast and efficient online publication tools to the scientific community, while keeping that community in charge of every aspect of the peer-review and publication process in order to ensure the highest quality standards in the journal.
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