CAREER: Topology and Geometry in Condensed Matter Systems

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

NONTECHNICAL SUMMARY This CAREER award supports theoretical research and education in the rapidly developing field of topological materials. The discovery of topological phases of matter is one of the most transformative recent breakthroughs in condensed matter physics, revealing new conceptual surprises in established topics such as the phases of matter and the behavior of electrons in insulators. Mathematically, topology refers to a property that remains unchanged when a sample is distorted in some way. Topologically nontrivial materials exhibit metallic surface states that are present regardless of how dirty the system is. However, from a practical perspective, the promise of devices harnessing these topological effects remains--for the most part--unrealized, due both to the lack of tools for finding realistic topological systems and the need for an improved understanding of the response of topological systems to external probes. One tool that can be leveraged to address these issues is geometry. Geometry enters into the description of crystal symmetry (example: an equilateral triangle looks the same after rotation by 60 degrees) and places constraints on the behavior of materials in the presence of electromagnetic fields and strain. The focus of this research is to use the interplay of geometry and topology to develop new insights into topological materials. The PI will use symmetry principles to determine ways to characterize topological materials through their behavior in external fields. Additionally, the interplay between crystal symmetries and electron-electron interactions in topological materials will be investigated with the goal of enabling the discovery of the next generation of topological materials. A major part of this research is directly applicable to ongoing work in experimental research laboratories. This research is closely integrated into an education plan at the undergraduate and graduate levels involving 1) the development of an advanced-level graduate course on Berry phases and topology in electronic structure, which is not currently covered in detail in current course offerings, 2) mentoring of undergraduate students in research both over the summer and during the school year, and 3) an outreach plan using techniques from statistical physics to study online harassment, in order to demonstrate the applications of physics to data science. TECHNICAL SUMMARY This CAREER award supports research into the interplay between crystal geometry and topological phenomena in order to develop a deeper understanding of quantum matter. The discovery of topological materials has revolutionized the understanding of quantum matter, demonstrating that not all insulators are created equal. The most striking experimental feature of topological materials is the existence of protected edge states, leading to protected non-dissipative conduction. However, topological materials also host remarkable bulk properties, such as non-dissipative transport coefficients, lack of localized electronic orbitals, and counterintuitive coupling to crystal geometry. The central goal of this award is to apply geometric data to compute previously unstudied properties of topological systems. This will be achieved through the study of 1. Response as a probe of topology: geometric transport coefficients such as the Hall viscosity will be studied to elucidate the interplay between anisotropy, geometric, and hydrodynamic response in topological systems. The proposed work will also determine the connection between topology and nonlinear electromagnetic response in a variety of experimentally relevant systems. 2. Role of symmetry in free-fermion band topology: the mathematical underpinnings of topological band theory will be extended in order better to understand the role of crystal symmetry in allowing for topologically nontrivial bands. Furthermore, symmetry principles will be applied to incommensurate and quasiperiodic structures to develop the theory of topological phases in quasiperiodic systems. 3. Crystal symmetries in interacting topological systems: the constraints of crystal symmetries will be incorporated into the study of many-body topological phases, and the theory of band representations for the elementary excitations in correlated phases will be developed. This CAREER project will serve to provide a richer understanding of topological and strongly correlated phases of quantum matter. A major part of this research is directly applicable to ongoing work in experimental research laboratories. This research is closely integrated into an education plan at the undergraduate and graduate levels involving 1) the development of an advanced-level graduate course on Berry phases and topology in electronic structure, which is not currently covered in detail in current course offerings, 2) mentoring of undergraduate students in research both over the summer and during the school year, and 3) an outreach plan using techniques from statistical physics to study online harassment, in order to demonstrate the applications of physics to data science. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.