Topological phases of matter and disordered systems

该奖项支持理论研究和教育，以研究和理解物质的性质和可能的相，特别是材料和工程材料结构中由电子组成的物质，在不同条件下。物质的拓扑相在理论上可以出现在各种材料和材料系统中。重要的例子包括在高磁场的工程半导体界面中捕获的二维电子液体，被称为拓扑绝缘体的特定绝缘体，以及某些类型的超导体。超导体是某些材料中的一种电子状态，具有独特的量子力学特性，可以不损失地传输电力。拓扑相的特点是具有坚固的特性，因此它们不容易被缺陷和缺陷破坏，但它们不像传统相那样具有对称性。理解拓扑相需要量子力学理论的基本方法。实验已经发现或证实了一些拓扑相的存在。这些物质的相有很大的潜力来引导或促进新技术，包括自旋电子器件，它不仅利用电子的电荷，而且利用其固有的磁性来操作，以及未来基于量子力学状态操纵的计算机技术。PI将以他之前的工作为基础，进一步发展这些相的理论，做出可以在实验中测试的理论预测，或者使识别拓扑相成为可能。具体的重点领域包括将拓扑相建模为网络的方法，以及在这些系统中出现的不寻常形式的粘度。在物质的其他相中，无序或固体中特有的随机杂质或缺陷的影响可以再次导致物质的不同相。这个项目将推进对这些的理论理解，特别是“旋转玻璃”。在这些系统的任何一个给定的例子中，很难猜测出最低的能量状态是什么，因为有许多低能量的替代品。量化这些数字是一个研究课题。从根本上说，这些玻璃系统与网络的许多其他应用以及计算机编程中的优化有关。该奖项支持拓扑相和无序材料的理论研究和教育。将考虑拓扑阶段的基本方面，以及应用和诊断工具。PI将在拓扑相的背景下研究纠缠。PI计划继续研究物质的手性拓扑相的张量网络态；张量网络在诸如一维拓扑相分类等方面取得了丰硕的成果。他的目标是研究与基本拓扑性质（如“中心电荷”）相关的输运现象问题，以及测量量子霍尔系统中霍尔粘度的可能性。关于无序系统：PI将使用复制对称破缺理论，在短期Ising自旋玻璃模型和其他模型中，得出低温下自旋玻璃相纯态数量的结果。这涉及到自旋玻璃理论中关于纯态的数量和组织的基本概念问题，是一个有争议的主题。

NONTECHNICAL SUMMARYThis award supports theoretical research and education to study and understand the properties and possible phases of matter, particularly matter made of electrons in materials and in engineered materials structures, under different conditions. Topological phases of matter are theoretically expected to occur in various materials and materials systems. Important examples include a two-dimensional liquid of electrons trapped at engineered semiconductor interfaces in a high magnetic field, specific insulators known as topological insulators, and in some types of superconductors. Superconductors are a state of electrons in some materials that have distinctive quantum mechanical properties and can transport electricity without loss. Topological phases are characterized by properties that are robust, so they cannot be easily destroyed by imperfections and defects, but they are not connected with symmetries as would be the case in more traditional phases. Understanding topological phases requires the theory of quantum mechanics in an essential way. Experiments have discovered or confirmed the existence of some topological phases. There is great potential for these phases of matter to lead or contribute to new technologies, including spintronic devices which exploit not only on the electron's charge but also its intrinsic magnetic properties for their operation, and in future computer technologies based on the manipulation of quantum mechanical states. The PI will build on his previous work and further develop the theory of these phases, making theoretical predictions that can be tested in experiments or make it possible to identify topological phases as they occur. The specific areas of focus include methods for modeling topological phases as networks, and an unusual form of viscosity that arises in some of these systems.In other phases of matter, disorder, or the effects of random impurities or defects that are endemic in solids, can lead again to distinct phases of matter. This project will advance the theoretical understanding of these, especially "spin glasses." In any given example of one of these systems it is very difficult to guess what the lowest energy state is, because there are many low-energy alternatives. Quantifying the number of these is a topic of research. In a fundamental way, these glassy systems are related to many other applications of networks, and to optimization in computer programming. TECHNICAL SUMMARYThis award supports theoretical research and education on topological phases and disordered materials. Fundamental aspects of topological phases will be considered, as well as applications and diagnostic tools. The PI will investigate entanglement in the context of topological phases. The PI plans to continue work on tensor network states for chiral topological phases of matter; tensor networks have been fruitful in, for example, classifying topological phases in one dimension. He aims to engage the problem of transport phenomena that are related to basic topological properties such as the "central charge," and the possibility of measuring the Hall viscosity in quantum Hall systems. On disordered systems: the PI will use replica symmetry breaking theory to produce results on the number of pure states in the spin-glass phase at low temperatures in the short-range Ising spin glass model, and others. This relates to basic conceptual issues in spin glass theory concerning the number and organization of pure states, and is a subject of controversy.