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.
