Free-particle descriptions of topological quantum matter and many-body localisation

自由粒子的概念是理论物理学的核心。这个简单的概念使我们能够描述自然界中各种各样的系统：例如，我们将原子解释为自由电子的集合，将电磁辐射解释为一组自由谐振子。但在现实世界中，粒子之间也会相互作用。相互作用在量子系统中有特别显著的影响，它们会导致远程相关和量子纠缠。这使得量子多粒子系统的理论描述非常具有挑战性。与此同时，人们对利用互动作为一种可能彻底改变技术的资源越来越感兴趣。现代科技一直以半导体、超导体和磁体等量子材料为基础，它们构成了激光器、晶体管、计算机等的基石。近年来，通过利用量子相关和量子纠缠，人们开始转向利用量子物理的全部力量，使这些设备变得更强大。因此，在这个阶段，有一个令人信服的理论和实践需要理解和操纵量子系统中相互作用的影响。本提案将研究两种当前感兴趣的物理系统，其中相互作用导致新的物理现象：(1)物质的拓扑相，包括马约拉纳和对偶费米子自旋链，自旋液体和分数量子霍尔态。这些相的形成是相互作用的“非微扰”效应的结果，不能被描述为自由电子的集合。这赋予了它们独特的性质，比如在任意但足够弱的扰动下具有鲁棒性。由于这种特殊的刚性，拓扑量子物质正被用作更强大的量子技术的基石，旨在适应环境扰动。(2)量子多粒子系统的非平衡动力学和热化。典型的量子系统是遍历的：它们很快达到热平衡，因为它们的组成粒子之间的相互作用很快消除了系统初始条件的记忆。然而，最近关于“多体定域”的研究表明，存在大量的强无序、相互作用的量子系统，它们无法达到热平衡。因此，这些系统是非遍历的，这意味着其中的量子效应可以持续异常长的时间，从而为量子技术提供了另一种保护途径。在本提案中，我们将开发一种新的方法来描述强相关现象中的相互作用效应，包括物质的拓扑相和多体局部化。我们将推进多体系统的建模在随机环境中使用最先进的数值模拟。我们对量子物质中拓扑和多体定域效应的理论研究将影响到冷原子、俘获离子、固体缺陷等方面的实验。最后，我们将探索在随机环境中实现具有拓扑顺序的相位的可能性，并提出具有增强抗热化稳定性的量子信息存储和处理方案。

The notion of a free particle is at the heart of theoretical physics. This simple notion allows us to describe a wide variety of systems in nature: for example, we explain atoms as collections of free electrons, and electromagnetic radiation as a set of free harmonic oscillators. But in real world, particles also interact with one another. Interactions have particularly striking effects in quantum systems, where they lead to long-range correlations and quantum entanglement. This makes the theoretical description of quantum many-particle systems very challenging. At the same time, there is growing interest in using interactions as a resource that could revolutionise technology. Modern technology has been based on quantum materials such as semiconductors, superconductors and magnets, which form the building blocks of lasers, transistors, computers, etc. In recent years, there has been a shift towards making such devices more powerful by exploiting the full power of quantum physics, which will be achieved by utilising quantum correlations and entanglement. Thus, at this stage, there is a compelling theoretical and practical need to understand and manipulate the effects of interactions in quantum systems.This proposal will investigate two physical systems of current interest where interactions lead to novel physical phenomena: (1) Topological phases of matter, which include Majorana and parafermion spin chains, spin liquids and fractional quantum Hall states. These phases form as a result of "non-perturbative" effects of interactions, and cannot be described as a collection of free electrons. This gives them unique properties such as robustness under arbitrary, but sufficiently weak, perturbations. Because of this special rigidity, topological quantum matter is being used as a building block of more robust quantum technologies, designed to be resilient to environmental perturbations.(2) Non-equilibrium dynamics and thermalisation in quantum many-particle systems. Typical quantum systems are ergodic: they quickly reach thermal equilibrium because the interactions between their constituent particles quickly erase the memory of the system's initial condition. However, recent work on "many-body localisation" shows that there exist large classes of strongly-disordered, interacting quantum systems which fail to reach thermal equilibrium. These systems are thus non-ergodic, which means that quantum effects in them can persist for unusually long times, thus providing another route of protection for quantum technology.In this proposal we will develop a new approach to describe interaction effects in strongly-correlated phenomena including topological phases of matter and many-body localisation. We will advance the modelling of many-body systems in random environments using state-of-the-art numerical simulations. Our theoretical investigation on the effects of topology and many-body localisation in quantum matter will impact several experiments on cold atoms, trapped ions, defects in solids, etc. Finally, we will explore the possibility of realising phases with topological order in random environments, and propose schemes for quantum information storage and processing with an enhanced stability against thermalisation.