Topology and Topological Phases in Strongly Interacting Many-Body Systems

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

TECHNICAL SUMMARY The Division of Materials Research and the Mathematical Sciences Division contribute funds to this award. It provides support for theoretical research and education on strongly interacting condensed matter many-body systems. The principal objective is to gain insight into the origin and behavior of topological phases. These phases are states of matter that are not the product of spontaneous symmetry breaking, and therefore do not have an order-parameter, but possess instead ?quantum order? in which the ground state degeneracy is determined by the topology of the space in which they live. Such topological phases have potential applications in quantum computing where the massive parallelism inherent in quantum time evolution provides fast solutions for problems that would require exponential time on conventional machines. The PI will further explore the correspondence between Calogero-Sutherland models and quantum Hall fluids with an aim to understanding the relation between edge and bulk states in non-abelian quantum fluids. The PI will explore new ideas for how to directly observe anionic statistics in the abelian quantum Hall effect through experiments. Research will also extend to the interplay of curvature and electronic structure in graphene. The specific aims of the research are three-fold: to understand how the local properties of the many-body wave-functions conspire to produce the global ground-state degeneracy that characterizes these systems, to explore possible applications of these systems for the construction of quantum computation devices, and to investigate novel systems that may, by design or by suitable engineering, possess the desired topological properties. Quantum field theory of many-body systems will be used to carry the research along with the representation theory of infinite dimensional current algebras, and index theorems that, under suitable circumstances, guarantee the existence of solutions of differential equations with desirable properties. This project contributes to the long-term goal of building a quantum computer, but also provides training of graduate students. These students will integrate research and education by acquiring valuable mathematical and computational skills, along with physical insight, that will prepare them for research careers in academia, industry, and the national laboratories. NON-TECHNICAL SUMMARY The Division of Materials Research and the Mathematical Sciences Division contribute funds to this award. It provides support for theoretical research and education at the interface of condensed matter physics and mathematics. The PI will use advanced theoretical methods, and mathematical and physical concepts to explore the notion of a new kind of spontaneous organization that appears in quantum mechanical systems. Well known is the notion of ordered states like the spontaneous organization of atoms in a regular array to form perfect crystalline materials. The ordered crystal state has a lower symmetry than the melt from which the crystal grows. In quantum mechanical systems, a new kind of order is possible in which there is no change in symmetry upon going to the organized state. Rather, the organization is reflected more abstractly in the topological properties of the quantum mechanical wavefunction. This notion became apparent from the study of a gas of electrons confined to a plane in a perpendicular magnetic field. The PI will use advanced theoretical techniques and mathematical concepts to further study this idea and to determine if a new topological quantum state of matter appears and can be directly observed in experiments. The exciting possibility is that this state of matter can be manipulated to perform computational operations that are intrinsically parallel and can execute certain algorithms at much higher speed than existing and foreseeable supercomputers. This research contributes to the intellectual foundations of quantum computing and toward its experimental realization. This award also supports education. Students will integrate research and education by acquiring valuable mathematical and computational skills, along with physical insight, that will prepare them to join the workforce of the 21st century.