Non-Perturbative Approaches to Condensed-Matter Physics

技术总结数学科学部和材料研究部为该奖项提供资金。它支持对具有强关联性的凝聚态系统的理论研究和教育，利用复杂的数学方法。一个主要主题是发现和理解具有拓扑有序的系统。拓扑有序存在于二维电子气中。在三维拓扑绝缘子中出现了一个密切相关的相。实现拓扑有序的另一个有希望的可能性是在自旋液体中。PI的目的是得到具有非阿贝尔统计的准粒子的准确结果，其中物理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格子模型。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打算利用对称性来实现精确计算。一种很有前途的方法是研究同时具有可积性和超对称性的模型；这些模型允许使用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式。这项研究处于数学和凝聚态物理的交界处。预测和发现的现象以及物质的新状态越来越需要更复杂的数学来描述它们。PI将继续与数学家合作；其结果可能导致凝聚态物理的进步和数学的新想法，特别是在拓扑学、组合学和分析方面。非技术总结数学科学部和材料研究部为该奖项提供资金。它支持在凝聚态理论中激起当前问题的研究和教育，这些问题正在导致物质的新状态的发现。PI将专注于物质的新状态，这些状态扩展了我们对组成物质的可区分阶段的理解。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层次的理解。其中包括最近发现的坚固的金属状态，它们出现在一类绝缘材料的表面，尽管这种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性质；其中之一是能够携带电流而不耗散。PI努力更深入地了解另一种物质状态的性质，这种物质出现在人造半导体材料中限制在二维电子中，并置于高磁场中。在适当的条件下，强相互作用的电子预计会出现一种新的物质状态。令人惊讶的是，这种状态似乎是由带电粒子组成的，每个粒子的电荷都等于电子电荷的一小部分。有人提出，对这些局限于二维的分数电荷“粒子”的操纵，可以使一种新的计算机技术成为可能，这种技术基于量子力学，在一些重要问题上的速度比现有计算机快许多倍。这项研究位于数学和凝聚态物理的交界处。预测和发现的现象以及物质的新状态越来越需要更复杂的数学来描述它们。PI将继续与数学家合作；其结果可能导致凝聚态物理的进步和数学的新想法，特别是在拓扑学、组合学和分析方面。

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.