Sierant Piotr, Lewenstein Maciej, Scardicchio Antonello, Vidmar Lev, Zakrzewski Jakub
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain.
ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain.
Rep Prog Phys. 2025 Jan 20;88(2). doi: 10.1088/1361-6633/ad9756.
Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBLgive rise to an MBL, in which the thermalization does not occur even in thelimit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.
统计力学提供了一个框架，用于描述大型、复杂多体系统的物理性质，仅使用几个宏观参数来确定系统的状态。对于孤立的量子多体系统，这种描述是通过本征态热化假设（ETH）实现的，该假设将热化、遍历性和量子混沌行为联系起来。然而，在强无序相互作用多体系统动力学中通过数值和实验发现的稳健多体局域化（MBL） regime 中，在有限的系统尺寸和演化时间内未观察到热化趋势。尽管MBL regime的现象学已得到充分确立，但核心问题仍然未得到解答：在什么条件下MBL会导致一种即使在无限系统尺寸和演化时间极限下也不会发生热化的MBL？本综述聚焦于近期旨在阐明MBL相状态的数值研究，并确立了关于无序多体系统动力学的关键开放性问题。过去几十年的研究带来了前所未有的各种新工具和指标来研究遍历性的破坏，从光谱和波函数测量、可观测量的矩阵元，到探测幺正量子动力学的量，再到输运和量子信息测量。我们对这些方法进行了全面概述，并试图对其主要特征提供统一理解。我们强调随着长度和时间尺度增加走向遍历性的一般趋势，这排除了简单的单参数标度假设，需要使用更精细的标度程序，并阻止将数值结果明确外推到渐近极限。在简要描述用于研究ETH和MBL的数值方法后，我们探索了各种解决MBL相问题的方法。在无序多体系统的本征值和本征向量导出的量中，持续出现朝向遍历性的有限尺寸漂移。这些漂移与多体系统动力学中观察到的朝向遍历性的连续缓慢变化和非零输运有关，即使在强无序情况下也是如此。这些现象阻碍了对ETH-MBL交叉处微观过程的理解。然而，随着无序强度增加动力学的突然减慢提供了暗示MBL相接近的前提。本综述得出结论，关于无序多体系统中的热化及其失效的问题仍然是一个引人入胜的有待进一步探索的开放领域。
