Non-Perturbative Approaches to Condensed-Matter Physics

数学科学部和材料研究部为该奖项提供资金。它支持理论研究和教育到凝聚态系统与强关联，利用数学复杂的方法。一个主题是发现和理解系统的拓扑秩序。拓扑秩序存在于二维电子气体。一个密切相关的阶段发生在三维拓扑绝缘体。 另一个有希望实现拓扑有序的可能性是自旋液体。PI的目的是推导出具有非阿贝尔统计的准粒子的精确结果，其中物理学取决于准粒子交换的顺序。PI还试图通过利用与二维经典可积模型的密切联系来寻找具有拓扑有序的晶格模型。另一个主题是开发新的数学工具来研究拓扑有序和其他新的物质状态。PI打算利用对称性，使精确计算成为可能。一个有前途的方法是研究具有可积性和超对称性的模型;这些模型允许使用来自不同数学领域的数学结果，如组合学，拓扑学和非线性微分方程。预测和发现的现象和新的物质状态越来越需要更复杂的数学来描述。PI将继续与数学家合作;其结果可能会导致凝聚态物理学的进步和数学的新思想，特别是在拓扑学，组合学和分析。非技术总结数学科学部和材料研究部为该奖项提供资金。它支持凝聚态理论中令人兴奋的当前问题的研究和教育，这些问题导致新的物质状态的发现。PI将专注于新的物质状态，扩展我们对物质可区分阶段的理解。拓扑学领域提供了数学思想，与量子力学一起导致对一些新的物质状态的更深入理解。这些包括最近发现的出现在一类绝缘材料表面的坚固金属状态，即使材料的大部分仍然是绝缘体。这些“拓扑绝缘体”的金属状态具有独特的特性;其中之一是能够携带电流而不耗散。PI致力于更深入地了解另一种物质状态的性质，这种物质状态出现在人工半导体材料中被限制在两个维度上的电子中，并置于高磁场中。在适当的条件下，一种新的物质状态被预测会从强相互作用的电子中出现。令人惊讶的是，这种状态似乎由带电粒子组成，每个粒子的电荷等于电子电荷的一部分。有人提出，操纵这些局限于二维空间的分数带电“粒子”，可以实现一种基于量子力学的新型计算机技术，在解决某些重要问题时比现有计算机快许多倍。这项研究处于数学和凝聚态物理学的交界处。预测和发现的现象和新的物质状态越来越需要更复杂的数学来描述。PI将继续与数学家合作;其结果可能会导致凝聚态物理学的进步和数学的新思想，特别是在拓扑学，组合学和分析。

TECHNICAL SUMMARYThe Division of Mathematical Sciences and the Division of Materials Research contribute funds to this award. It supports theoretical research and education into condensed matter systems with strong correlations, utilizing mathematically sophisticated methods.A main theme is to find and understand systems with topological order.Topological order exists in two-dimensional electron gases. A closely related phase occurs in three-dimensional topological insulators. Another promising possibility for realizing topological order is in spin liquids. The PI aims to derive exact results for quasiparticles with non-Abelian statistics, where the physics depends on the order in which the quasiparticles are interchanged. The PI also seeks to find lattice models with topological order by exploiting the close connection to two-dimensional classical integrable models.Another theme is to develop new mathematical tools to study topologically ordered and other new states of matter. The PI intends to exploit symmetries that will make exact computations possible. One promising approach is to study models with both integrability and supersymmetry; these models allow the use of mathematical results from disparate areas of mathematics such as combinatorics, topology, and non-linear differential equations.This research lies at an interface of mathematics and condensed matter physics. Predicted and discovered phenomena and new states of matter increasingly require more sophisticated mathematics for their description. The PI will continue to collaborate with mathematicians; the result may lead to advances in condensed matter physics and to new ideas in mathematics, particularly in topology, combinatorics and analysis.NON-TECHNICAL SUMMARYThe Division of Mathematical Sciences and the Division of Materials Research contribute funds to this award. It supports research and education in exciting current problems in condensed-matter theory which are leading to discoveries of new states of matter. The PI will focus on new states of matter that extend our understanding of what constitutes distinguishable phases of matter. The field of topology provides mathematical ideas that together with quantum mechanics lead to deeper understanding of some new states of matter. These include recently discovered robust metallic states that appear on the surface of a class of insulating materials, eventhough the bulk of the material remains an insulator. The metallic states of these "topological insulators" have distinctive properties; among them is the ability to carry electric currents without dissipation. The PI strives for a deeper understanding of the nature of another state of matter that arises in electrons confined to two dimensions in artificial semiconductor materials and placed in a high magnetic field. Under appropriate conditions, a new state of matter is predicted to emerge from the strongly interacting electrons. Amazingly, this state appears to consist of charged particles, each one having a charge equal to a fraction of an electron charge. It has been proposed that manipulation of these fractionally charged "particles" confined to two dimensions can enable a new kind of computer technology that is based on quantum mechanics and is many times faster than existing computers for some important kinds of problems.This research lies at an interface of mathematics and condensed matter physics. Predicted and discovered phenomena and new states of matter increasingly require more sophisticated mathematics for their description. The PI will continue to collaborate with mathematicians; the result may lead to advances in condensed matter physics and to new ideas in mathematics, particularly in topology, combinatorics and analysis.