Topological and Disordered Phases of Matter

该奖项支持对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都有一个电子，并且它们可以在低温下保持活性。在电子的量子力学世界中，由于电子之间的相互作用导致了它们运动中的相关性，新的现象可能会出现。它们还可以导致形成一种电子物质的状态，在我们熟悉的由经典力学控制的世界中没有类似的东西。这种状态的一个特征是量子纠缠，这意味着电子之间存在非常长距离的相关性。被研究的物质的量子相，即物质的拓扑相，涉及到非平凡的长程纠缠。理论上研究这种物质态的各种方法是已知的。其中一些涉及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分支代数拓扑学的方法。PI将把他的经验与这些承担的问题。对这些主题的研究可能有助于量子信息科学，其中有利用物质的拓扑相来执行量子计算的建议。这些不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中。PI还将研究另一个独特的研究领域，无序系统。这意味着自由度，例如玻璃中的原子，彼此相互作用的强度在不同的地方不同，并且被建模为随机数。这一方面，它模拟了真实的材料固有的完美性的缺乏，导致了一些非常困难的问题;一类特殊的例子被称为“自旋玻璃”。找到最低的能量状态可能是一个计算困难的问题。即使在统计水平上，也很难描述自旋玻璃的性质。PI将继续使用严格的数学方法来促进对这一问题的理解。该奖项支持理论研究和教育，研究经典和量子凝聚态系统（通常是晶格系统）中的低温现象。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。一种特别感兴趣的方法是张量网络状态。PI将研究这些方法是否可以成功地应用于多个维度的物质拓扑相，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，量子信息理论，以及多体和量子场论。PI还将研究无序系统，如经典的自旋玻璃。突出的有争议的问题包括一个基本问题，即在一个无限系统的极限中，在低温相是否有许多纯态;这就是复制对称性破缺的问题。使用严格的数学分析可能是唯一的方法，可以回避有关解释的争议，一直困扰着非严格和计算的方法。PI使用纽曼和斯坦的元态方法来提供一个严格的框架来解决问题，但也可能使用复制方法来获得洞察力，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，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响评审标准。

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.