Geometric and Topological Phenomena in Condensed Matter Systems

该奖项支持拓扑绝缘体的理论研究和教育。PI将显示晶体对称性是否会影响体带十倍分类的细化。大部分的研究工作将集中在探索拓扑绝缘体的体面连接与它们的粒子和引力物理类似物之间的类比。特别是，固态系统提出了思考量子场论中手性和规范异常的新方法。这将导致另一种计算夸克-胶子等离子体输运系数的方法。特别地，我们将探讨异常多项式的系数与那些出现在著名的索默菲尔德展开中的系数之间的一些奇怪的关系。这项探索的一个关键因素是开发了一套工具，用于描述量子自旋作为经典力学系统的动力学和量子拓扑。这项研究将表明这些“经典”工具是否可以扩展到导出引力异常的固态版本，该版本可用于计算量子霍尔效应的热模拟系数。拟议的研究计划将通过为构建可能导致量子计算机的未来一代固态电子设备提供指导方针，并通过允许以比建造大型加速器低得多的成本探索推测粒子物理，从而影响理论凝聚态物理领域以外的领域。PI将开发一个门户网站，让学生和公众了解凝聚态物理的最新发展。该奖项支持拓扑绝缘体的理论研究和教育。对宇宙的完整理解可以用粒子物理学的“标准模型”来概括。在这个模型中，物质的基本部分通过规范场相互作用，规范场是电和磁的概括。其中一些相互作用，即“弱”力，具有一个奇怪的特征，即它们只影响物质的左旋部分。在数学上，产生这种手性的自然方式是物质被困在嵌入某些高维空间的表面上。这一观点听起来像是科幻小说里的东西，但近年来的实际研究表明，某些含有重元素的晶体绝缘体正是以这种方式表现出来的。这些“拓扑绝缘体”只是它们内部的绝缘体。在它们的表面，电子可以自由移动和导电，但它们这样做的方式与夸克相互作用的方式形成了精确的类比。这些导电态引起了极大的兴趣，部分原因是由于建造新一代固态电子设备的实际原因，这些设备可能会导致量子计算机，但也因为与粒子物理学的类比允许以比建造大型加速器低得多的成本探索推测性粒子物理学。PI将开发一个门户网站，让学生和公众了解凝聚态物理的最新发展。

Technical SummaryThis award supports theoretical research and education on topological insulators. The PI will show whether crystal symmetries effect a refinement of the ten-fold way classification of the bulk bands. A large portion of the research efforts will be directed at exploring the analogies between the bulk-surface connection in topological insulators and their particle and gravitational physics analogs. In particular, the solid-state systems suggest new ways of thinking about chiral and gauge anomalies in quantum field theory. This will lead to an alternative method for calculation of transport coefficients in quark-gluon plasmas. In particular some curious relations between the coefficients in anomaly polynomials and those appearing in the well-known Sommerfeld's expansion will be explored. A key ingredient in this exploration is a set of tools developed for the description of the dynamics and quantum topology of a quantum spin as a classical mechanical system. This study will show whether these "classical" tools can be extended to derive the solid-state version of the gravitational anomaly which can be used to calculate the coefficient of the thermal analogue of the quantum Hall effect. The proposed research program will have impact beyond the field of theoretical condensed matter physics by providing guidelines for building a future generation of solid-state electronic devices that might lead to quantum computers and by allowing exploration of the speculative particle physics at a much lower cost than building large accelerators. The PI will develop a web portal to expose students and the public to recent developments in condensed matter physics. Non-Technical Summary This award supports theoretical research and education on topological insulators. A complete understanding of the universe is summarized in the "standard model" of particle physics. In this model the fundamental bits of matter interact through gauge fields which are generalizations of electricity and magnetism. Some of these interactions, the "weak" force, possess the curious feature that they only affect the left-handed parts of matter. A mathematically natural way for this handedness to arise is for matter to be trapped on a surface embedded in some higher dimensional space. This notion sounds as if it belongs in a science fiction story, but in recent years quite down-to-earth studies have shown that certain crystalline insulators containing heavy elements behave in exactly this way. These "topological insulators" are only insulators on their insides. On their surfaces, electrons are free to move and conduct electricity, but they do so in a manner that forms a precise analogue of the way that quarks interact. These conducting states are of great interest, partly for practical reasons in building new generations of solid-state electronic devices that might lead to quantum computers, but also because the analogy with particle physics allows exploration of speculative particle physics at a much lower cost than building large accelerators. The PI will develop a web portal to expose students and the public to recent developments in condensed matter physics.