CAREER: Strong Disorder and Electron Interaction Effects in Topological Insulators

材料研究部和数学科学部为该职业奖提供资金，以支持对拓扑绝缘体的研究，拓扑绝缘体是一类具有有趣且潜在有用的边缘和表面物理的新材料。表面和边缘状态源于材料本体内部量子态的拓扑性质。因此，预期边缘和表面的性质对于降解、变形或化学污染是稳健的。然而，它们可以被大块结构的大而平滑的变形或强烈的无序和电子相互作用所破坏。PI的目的是量化拓扑绝缘体可以支持多少无序而不失去其确定的属性，并了解电子-电子相互作用的影响。当无序和相互作用很强时，传统理论方法的洞察力是有限的。PI将开发新的分析方法。具体而言，PI将：1）利用非交换几何的方法定义了强无序和电子-电子相互作用下的鲁棒拓扑不变量，并给出了保证它们量子化和不变性的精确条件，2）设计数值算法来实现非交换演算，并进行明确的计算机模拟，以绘制出各种现有的拓扑绝缘体在强3）利用1和2的结果，预测和表征具有新拓扑性质的材料。该提案的教育部分将通过一系列有计划的、深入的和教学性的评论以及座谈会类型的出版物，促进在广大科学界传播理论凝聚态物理学中的现代问题和现代数学分析方法。PI将发起“凝聚态物质黑板讲座”，这是一个加强纽约市大都市地区凝聚态物质和数学物理学家之间交流和合作的论坛。此外，还计划在PI的机构-叶史瓦大学斯特恩女子学院-开展一系列活动并提供研究奖学金，这将提高代表性不足的少数民族学生对科学的参与，特别是在理论凝聚态物理学领域。材料研究部和数学科学部为该职业奖提供资金，以支持对新发现的称为拓扑绝缘状态的物质状态的研究。表现出这种状态的材料具有非常不寻常的性质，例如，虽然材料的主体是电绝缘体，但沿着切割到材料主体中的任何边缘或表面都存在完美的导电电荷和自旋通道。这些边缘或表面导电通道是坚固的，也就是说，它们在适度的机械、化学或热应力下不会消失。这些材料可能有助于为未来的电子、计算机技术和清洁能源生产、运输和储存技术奠定基础。PI将专注于拓扑绝缘体的表征，当缺陷由于各种因素（如制造过程，高热，辐射或机械应力）而扩散到大量时。由于材料的结构将不再在原子水平上完全有序，而是随机扭曲，因此通常很难用现有的理论工具来表征或预测这些材料。为了克服这一困难，PI将采用一种称为“非交换微积分”的数学形式，这是传统微积分和几何的推广，适用于无法定义基本光滑空间或几何对象的情况。这将使人们能够更深入地了解拓扑绝缘体的特性。从这些研究中获得的知识将指导实验人员如何改善现有拓扑绝缘体的性能，并有助于搜索和发现具有令人兴奋的特性的新材料。该提案的教育部分将通过一系列有计划的、深入的和教学性的评论以及座谈会类型的出版物，促进在广大科学界传播理论凝聚态物理学中的现代问题和现代数学分析方法。PI将发起“凝聚态物质黑板讲座”，这是一个加强纽约市大都市地区凝聚态物质和数学物理学家之间交流和合作的论坛。此外，还计划在PI的机构-叶史瓦大学斯特恩女子学院-开展一系列活动并提供研究奖学金，这将提高代表性不足的少数民族学生对科学的参与，特别是在理论凝聚态物理学领域。

TECHNICAL SUMMARY The Division of Materials Research and the Division of Mathematical Sciences contribute funds to this CAREER award to support research on topological insulators, a new class of materials that have interesting and potentially useful edge and surface physics. The surface and edge states arise from topological properties of the quantum states inside the bulk of the material. As such, the properties of the edges and surfaces are expected to be robust against degradation, deformation or chemical contamination. They can, however, be destroyed by large, smooth deformations of the bulk structure or by strong disorder and electron interactions. The PI aims to quantify how much disorder a topological insulator can support without losing its definitive properties and to understand the effects of electron-electron interaction. The insight from traditional theoretical methods is limited when disorder and interactions are strong. The PI will develop new methods of analysis. Specifically, the PI will: 1) Use the methods of Non-Commutative Geometry to define robust topological invariants in the presence of strong disorder and electron-electron interaction, and to map out the precise conditions that assure their quantization and invariance, 2) Devise numerical algorithms to implement the non-commutative calculus and carry out explicit computer simulations to map out the phase diagram of various existing topological insulators in the presence of strong disorder, and 3) Using the results of 1 and 2, predict and characterize materials with novel topological properties. The educational component of the proposal will contribute to the dissemination of modern problems in theoretical condensed matter physics and of the methods of modern mathematical analysis among a broad scientific community, through a series of planned, in-depth, and pedagogical reviews as well as colloquium-type publications. The PI will initiate the "Condensed Matter Blackboard Lectures", a forum to enhance communication and collaboration between condensed matter and mathematical physicists in the metropolitan New York City area. Furthermore, a series of activities and research scholarships are planned at the PI's institution, Stern College for Women of Yeshiva University, which will enhance the participation of the underrepresented minority students in science, in particular, in the field of theoretical condensed matter physics. NON-TECHNICAL SUMMARY The Division of Materials Research and the Division of Mathematical Sciences contribute funds to this CAREER award to support research on a newly discovered state of matter called the topological insulating state. Materials exhibiting this state have highly unusual properties, e.g. while the bulk of the material is an electrical insulator, there are perfectly conducting charge and spin channels along any edge or surface that is cut into the bulk of the material. These edge or surface conducting channels are robust, that is, they do not vanish under moderate mechanical, chemical or heat stress. These materials may contribute to the foundations of future electronics, computer technology and clean energy generation, transportation and storage technologies. The PI will focus on the characterization of topological insulators when imperfections proliferate to large numbers due to various factors such as the fabrication process, intense heat, radiation or mechanical stress. Since the structure of the material will no longer be perfectly ordered at the atomic level, but rather be randomly distorted, it is generally very difficult to characterize or make predictions about these materials with existing theoretical tools. To overcome this difficulty, the PI will employ a mathematical formalism called "non-commutative calculus", which is a generalization of the traditional calculus and geometry to cases when no underlying smooth space or geometrical object can be defined. This will enable a deeper understanding of the properties of topological insulators. The knowledge gained from these studies will guide experimentalists on how to improve the performance of existing topological insulators and aid in the search and discovery of new materials with exciting properties. The educational component of the proposal will contribute to the dissemination of the modern problems in theoretical condensed matter physics and of the methods of modern mathematical analysis among a broad scientific community, through a series of planned, in-depth, and pedagogical reviews as well as colloquium-type publications. The PI will initiate the "Condensed Matter Blackboard Lectures", a forum to enhance communication and collaboration between condensed matter and mathematical physicists in the metropolitan New York City area. Furthermore, a series of activities and research scholarships are planned at the PI's institution, Stern College for Women of Yeshiva University, which will enhance the participation of the underrepresented minority students in science, in particular, in the field of theoretical condensed matter physics.