Exotic physics in quantum Hall and spin liquid systems

物质可能有哪些相？这是凝聚态物理的一个中心问题。历史上使用了两种组织原理：自由电子带理论和自发对称破缺理论。量子凝聚态物理的一个现代主题是超越这两种范式去理解相互作用的多体系统。两个突出的例子是分数量子霍尔效应和量子自旋液体。这些系统最显著的共同特征是电子分馏：在低温下，电子的行为就好像它们被“分裂”成更小的部分（比如带有分数电荷或统计数据的粒子）。这证明了涌现原理的力量：多体系统的集体行为可能与其微观组成部分截然不同。从现代观点来看，电子分馏最终来自远程量子纠缠。我的研究计划将集中在更好地理解由于远距离纠缠而产生的各种紧急现象，如分馏。在接下来的五年里，我想更好地理解(1)在给定的系统中允许的分馏模式是什么？(2)无序对分馏物理的影响是什么？最近的实验和理论发展很好地激发了这些问题。例如，问题(1)的自然背景是理解在一个特别有趣的系统中被称为5/2量子霍尔态的不同可能的分数化模式。这个问题的答案将有助于将5/2状态明确地建立为非阿贝尔拓扑顺序，这可能对未来的量子信息处理有用。问题(2)是由量子自旋液体的研究引起的：在过去的十年中，已经发现了许多“候选”自旋液体材料，但它们大多数都比较脏。因此，理解量子自旋液体中的无序效应对于将理论与实验联系起来至关重要。对多体系统中远程量子纠缠的研究不仅本身具有深刻的概念价值，而且还可以为未来的技术发展铺平道路，从量子信息处理到具有以前无法想象的电性能的材料——就像自由电子和对称破断理论为半导体、磁铁和超导体的物理学奠定基础一样。所有这些都在上个世纪的技术革命中发挥了重要作用。这个研究项目将培养大约四名博士生。他们将掌握现代物理（如量子场论、统计力学和固态物理）中的各种分析和数值技能。更重要的是，他们将学会像物理学家一样思考和交流。所有这些都将有利于他们未来的职业生涯，无论是在学术界还是在与技术相关的行业。

What are the possible phases of matter? This is a central question of condensed matter physics. Historically two organizing principles have been used: the free electron band theory and the theory of spontaneous symmetry breaking. A modern theme of quantum condensed matter physics is the quest to understand interacting many-body systems beyond these two paradigms. Two prominent examples are the fractional quantum Hall effect and quantum spin liquids. The most striking common feature of these systems is electron fractionalization: at low temperature the electrons behave as if they are "split" into smaller fractions (such as particles with fractional electric charge or statistics). This demonstrates the power of the principle of emergence: the collective behavior of a many-body system can be drastically different from its microscopic constituents. From a modern point of view electron fractionalization ultimately come from long-range quantum entanglement. My research program will be focused on better understanding various emergent phenomena due to long-range entanglement such as fractionalization.  In the next five years I would like to better understand (1) what are the allowed patterns of fractionalization in a given system? and (2) what are the effects of disorder on the physics of fractionalization? These questions are well motivated by recent experimental and theoretical developments. For example, a natural context for question (1) is to understand different possible patterns of fractionalization in a particularly interesting system known as the 5/2 quantum Hall state. An answer to this question would help establishing the 5/2 state unambiguously as a non-abelian topological order, potentially useful for future quantum information processing. Question (2) is motivated by the study of quantum spin liquids: during the past decade many "candidate" spin liquid materials have been discovered, but most of them tend to be rather dirty. An understanding of disorder effects in quantum spin liquids will therefore be crucial for bridging theories with experiments. The study of long-range quantum entanglement in many-body systems not only bears deep conceptual value on its own, but could also pave the way for future technological developments, ranging from quantum information processing to materials with previously unconceived electric properties - just like how the theories of free electrons and symmetry breaking laid the foundations for physics of semiconductors, magnets and superconductors, all of which played major roles in the technological revolution last century. This research program will train around four Ph.D. students. They will acquire various analytic and numerical skills in modern physics (such as those in quantum field theory, statistical mechanics and solid state physics). More importantly they will learn to think and communicate like physicists. All these will benefit their future career, in academia or in technology-related industry.