Non-perturbative and stochastic approaches to many-body localization

物理学的一个中心问题是任何给定系统的复杂性。表征系统行为的量的本质是什么？哪些特征可以安全地忽略？系统对外部扰动有多敏感？它的动力学有多可预测？需要控制多少参数才能以特定的方式操作系统？这些问题不仅是统计力学等基础领域的核心，也是平衡相和非平衡动力学表征的关键，并在量子信息、控制和传感等领域广泛应用。为了满足在量子环境中理解这些问题的迫切需要，科学界最近确定了一个应该包含许多所需答案的关键范式。这涉及具有无序、低空间维度和局部相互作用的大而有限的多体系统。这些成分被认为共同作用于初始条件的局部记忆的持久性，这是一种被称为多体定位（MBL）的迷人现象。对这一现象的理解仍处于初级阶段。MBL的理论论证是彻底的，但本质上也是定性的，而迄今为止的定量见解主要来自相对较小（在现实世界中）系统的数值研究。这种状态导致了基于输运、热化、纠缠、动力学等特征的激增，而这些特征的详细相互关系大多不清楚。这些缺点变得更加紧迫，因为第一个专门的实验针对的是另一组特征，即在现实世界系统中直接可观察到的后果。在这个项目中，我们退后一步，提出一个问题：什么样的理解水平可以被接受，以构成对多体本地化和非本地化的相当完整的描述？借鉴非相互作用系统的经验，我们认为这必须包括准一维设置中的统计描述，作为中心部分。我们通过（I.）源自晶格场理论的丰富一维模型系统（Schwinger模型，量子电动力学的一个版本）来实现这一目标，可以进行详细的处理；（II.）基于密度矩阵的基本组成规则的随机描述（描述纠缠的关键对象），允许全面访问一般系统行为。在可观察到的结果中，我们的目标是（III）区分MBL相和相变的一般和系统特定方面，包括实验可测试的特征，以及（IV）将考虑扩展到拓扑相，其中系统由于复杂的量子效应而显示有序。

A central question in physics concerns the complexity of any given system. What is the nature of quantities characterising the system behaviour, and which characteristics can be safely ignored? How sensitive is the system against external perturbations and how predictable is its dynamics? How many parameters need to be controlled to manipulate the system in a specific way? These questions not only lie at the heart of fundamental fields such as statistical mechanics, but also provide the key to the characterisation of equilibrium phases and nonequilibrium dynamics and pervade practical applications in quantum information, control and sensing.Meeting the urgent need to understand these questions in the quantum setting, the scientific community has recently identified a key paradigm that should hold many of the required answers. This concerns large but finite many-body systems with disorder, low spatial dimensionality and local interactions. These ingredients have been seen to conspire in a persistence of local memory of initial conditions, a fascinating phenomenon known as many-body localisation (MBL). The understanding of this phenomenon is still in its infancy. The theoretical arguments for MBL are thorough, but also in essence qualitative, while quantitative insight to date mainly arrives from numerical investigations of relatively small (in real-world terms) systems. This state of affairs has given rise to a proliferation of characterisations based, among others, on transport, thermalisation, entanglement, dynamics, whose detailed mutual relationships are mostly unclear. These shortcomings become even more pressing as the very first dedicated experiments target yet another set of characteristics, the directly observable consequences in real-world systems.In this project, we take a step back and ask the question: which level of understanding could be accepted to constitute a rather complete description of many-body localisation and delocalisation? Taking a leaf out of the book from non-interacting systems, we argue that this has to involve, as a central piece, a statistical description in quasi-one dimensional settings. We approach this goal from (I.) a rich one-dimensional model system originating from lattice field theory (the Schwinger model, a version of quantum electrodynamics) which is amenable to a detailed treatment and (II.) a stochastic description based on fundamental composition rules of the density matrix (the key object to describe entanglement), allowing comprehensive access to the generic system behaviour. Amongst the observable consequences, we aim to (III.) discriminate between general and system-specific aspects in the MBL phase and of the phase transition, including experimentally testable signatures, and (IV.) extend the considerations to topological phases, where the system displays order due to intricate quantum effects.