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Topological and Disordered Phases of Matter

物质的拓扑相和无序相

基本信息

批准号：
1724923
负责人：
Nicholas Read
金额：
$ 36万
依托单位：
Yale University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1724923&HistoricalAwards=false
关键词：
Topological Disordered Phases Matter

项目摘要

NONTECHNICAL SUMMARYThe award supports theoretical research and education on profound questions that arise in the study of modern materials. In a solid material, there are many electrons, say one for each atom in the material, and they may remain active at low temperatures. In the quantum mechanical world of the electron, new phenomena can arise as a consequence of interactions between electrons that lead to correlations in their motion. They can also lead to the formation of a state of electronic matter with no analog in the familiar world governed by classical mechanics. A feature of this state is quantum entanglement, meaning very long-range correlations between electrons. The quantum phases of matter under investigation, known as topological phases of matter, involve non-trivial long-range entanglement.Various ways of theoretically studying such states of matter are known. Some involve techniques called tensor networks. The PI will explore the intrinsic limitations of these methods. Other approaches involve methods from quantum field theory and from the branch of mathematics known as algebraic topology. The PI will bring his experience with these to bear on the problems. Research on these topics may contribute to quantum information science, where there are proposals that utilize topological phases of matter to perform quantum computation. There is also the possibility that these unusual states of matter could be applied as novel materials in other technological applications.The PI will also study another distinct area of research, disordered systems. This means that the degrees of freedom, for example atoms in glass, that interact with one another do so with strengths that differ from one place to another, and are modeled as random numbers. This aspect, which models the lack of perfection intrinsic to real materials, leads to some very difficult questions; a particular class of examples are called "spin glasses". Finding the lowest energy state may be a computationally hard problem. Even at a statistical level, it may be very hard to characterize the properties of a spin glass. The PI will continue the use of rigorous mathematical approaches to advance understanding of this problem. TECHNICAL SUMMARYThe award supports theoretical research and education to study low-temperature phenomena in both classical and quantum condensed matter systems, usually in lattice systems.On the quantum side, the goal of these studies is to understand the topological phases of matter in these systems in greater depth. An approach of particular interest is tensor network states. The PI will investigate whether these methods can be successfully applied to topological phases of matter in more than one dimension, or whether they possess intrinsic (topological) limitations that make these applications impossible. The theoretical techniques that will be used include concepts from algebraic topology, such as K-theory, and quantum information theory, as well as many-body and quantum-field theory. The PI will also study disordered systems such as classical spin glasses. Outstanding controversial issues include the basic question of whether or not there are many pure states in the low temperature phase in the limit of an infinite system; this is the question of replica symmetry breaking. The use of rigorous mathematical analysis may be the only approach that can sidestep disputes about interpretation that have dogged non-rigorous and computational approaches. The PI uses Newman and Stein's metastate approach to provide a rigorous framework for attacking the problem, but may also uses replica methods to gain insight and to make connections with non-rigorous methods or experiments.In both areas, while the emphasis is on fundamental theoretical issues, experimental relevance is always in view.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

非技术总结该奖项支持关于现代材料研究中出现的深刻问题的理论研究和教育。在固体材料中，有许多电子，比如说材料中的每个原子一个电子，它们在低温下可能保持活跃。在电子的量子力学世界中，电子之间的相互作用可能会产生新的现象，这些相互作用导致电子运动中的关联。它们还可以导致形成一种电子物质的状态，这在经典力学所支配的熟悉世界中是没有类似的。这种状态的一个特征是量子纠缠，这意味着电子之间的非常远距离的关联。被研究的物质的量子相，被称为物质的拓扑相，涉及到非平凡的长程纠缠。从理论上研究物质的这种状态的方法多种多样。其中一些涉及被称为张量网络的技术。PI将探索这些方法的内在局限性。其他方法涉及量子场论和数学的一个分支--代数拓扑学的方法。私家侦探将利用他在这些方面的经验来处理这些问题。对这些主题的研究可能有助于量子信息科学，在量子信息科学中，有提议利用物质的拓扑相来执行量子计算。这些不同寻常的物质状态也有可能作为新材料应用于其他技术应用。PI还将研究另一个不同的研究领域，无序系统。这意味着，相互作用的自由度，例如玻璃中的原子，其强度因地而异，并被建模为随机数。这一方面模拟了真实材料内在的缺乏完美性，导致了一些非常困难的问题；一类特殊的例子被称为“自旋眼镜”。寻找最低能量状态可能是一个计算困难的问题。即使在统计水平上，也可能很难描述自旋玻璃的性质。国际和平研究所将继续使用严格的数学方法来促进对这一问题的理解。技术总结该奖项支持理论研究和教育，以研究经典和量子凝聚物质系统中的低温现象，通常是在晶格系统中。在量子方面，这些研究的目标是更深入地了解这些系统中物质的拓扑相。一种特别有趣的方法是张量网络状态。PI将调查这些方法是否可以成功地应用于不止一个维度的物质的拓扑相，或者它们是否具有使这些应用不可能的内在(拓扑)限制。将使用的理论技术包括代数拓扑学的概念，如K理论和量子信息论，以及多体和量子场论。PI还将研究无序系统，如经典自旋玻璃。悬而未决的争议问题包括无限系统极限的低温阶段是否存在许多纯态的基本问题，这就是复制对称破缺问题。使用严格的数学分析可能是唯一可以回避关于解释的争议的方法，这些争议一直困扰着非严格的和计算的方法。PI使用Newman和Stein的转移方法来提供解决问题的严格框架，但也可能使用复制方法来获得洞察并与非严格的方法或实验建立联系。在这两个领域，虽然重点是基本的理论问题，但实验相关性始终存在。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。