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。表面态和边缘态是由材料内部量子态的拓扑特性产生的。因此，预计边缘和表面的性能对降解，变形或化学污染是坚固的。然而，它们可以被大块结构的大而平滑的变形或强无序和电子相互作用所破坏。PI旨在量化拓扑绝缘体在不失去其确定特性的情况下可以支持多少无序，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)利用非交换几何方法定义强无序和电子-电子相互作用下的鲁棒拓扑不变量，并绘制出保证其量子化和不变性的精确条件；2)设计数值算法实现非交换演算，并进行显式计算机模拟，绘制出强无序存在下现有各种拓扑绝缘体的相图；3)利用1和2的结果，预测和表征具有新型拓扑性质的材料。该提案的教育部分将通过一系列有计划的、深入的、教学性的评论以及座谈会式的出版物，在广泛的科学界传播理论凝聚态物理中的现代问题和现代数学分析方法。PI将发起“凝聚态黑板讲座”，这是一个论坛，旨在加强纽约大都会地区凝聚态和数学物理学家之间的交流和合作。此外，PI的机构，叶史瓦大学斯特恩女子学院计划开展一系列活动和研究奖学金，这将加强代表性不足的少数民族学生在科学领域的参与，特别是在理论凝聚态物理领域。材料研究部和数学科学部向CAREER奖提供资金，以支持对一种新发现的物质状态的研究，这种状态被称为拓扑绝缘状态。表现出这种状态的材料具有非常不寻常的特性，例如，虽然材料的主体是电绝缘体，但在材料主体的任何边缘或表面上都有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。这些材料可能为未来的电子、计算机技术和清洁能源的产生、运输和储存技术奠定基础。当由于制造工艺、强热、辐射或机械应力等各种因素导致缺陷大量增加时，PI将重点关注拓扑绝缘体的特征。由于材料的结构在原子水平上不再是完美有序的，而是随机扭曲的，用现有的理论工具来描述或预测这些材料通常是非常困难的。为了克服这个困难，PI将采用一种称为“非交换微积分”的数学形式，它是传统微积分和几何的推广，适用于无法定义底层光滑空间或几何对象的情况。这将使我们对拓扑绝缘体的性质有更深入的了解。从这些研究中获得的知识将指导实验学家如何改进现有拓扑绝缘体的性能，并帮助寻找和发现具有令人兴奋性能的新材料。该提案的教育部分将通过一系列有计划的、深入的、教学性的评论以及座谈会式的出版物，在广泛的科学界传播理论凝聚态物理中的现代问题和现代数学分析方法。PI将发起“凝聚态黑板讲座”，这是一个论坛，旨在加强纽约大都会地区凝聚态和数学物理学家之间的交流和合作。此外，PI的机构，叶史瓦大学斯特恩女子学院计划开展一系列活动和研究奖学金，这将加强代表性不足的少数民族学生在科学领域的参与，特别是在理论凝聚态物理领域。

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.