Non-perturbative and stochastic approaches to many-body localization

物理学中的一个中心问题涉及任何给定系统的复杂性。表征系统行为的量的性质是什么，哪些特征可以安全地忽略？系统对外部扰动的敏感度如何？其动态的可预测性如何？需要控制多少参数才能以特定的方式操纵系统？这些问题不仅是统计力学等基础领域的核心问题，也是描述平衡相和非平衡动力学的关键，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关键范式，它应该包含许多所需的答案。这涉及大但有限的多体系统与无序，低空间维度和局部相互作用。这些成分被认为是初始条件的局部记忆持续存在的阴谋，这是一种被称为多体定位（MBL）的迷人现象。对这一现象的认识仍处于初级阶段。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。这种状况已经引起了基于运输，热化，纠缠，动力学等的表征的扩散，其详细的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目标又一组特性，在现实世界的系统中的直接可观察的后果。在这个项目中，我们退一步，并提出这样的问题：哪一个层次的理解可以接受构成一个相当完整的描述多体定位和离域？从非相互作用系统的书中抽出一页，我们认为这必须涉及，作为一个核心部分，在准一维设置的统计描述。我们从（一）一个丰富的一维模型系统起源于晶格场理论（Schwinger模型，量子电动力学的一个版本），这是经得起详细的治疗和（II.）基于密度矩阵（描述纠缠的关键对象）的基本组成规则的随机描述，允许全面访问通用系统行为。在可观察到的结果中，我们的目标是（III.）区分MBL阶段和相变的一般和系统特定方面，包括实验可测试的签名，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出秩序。

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.