Non-Perturbative Approaches to Condensed-Matter Physics

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

TECHNICAL SUMMARY The Division of Mathematical Sciences and the Division of Materials Research contribute funds to this award. It supports theoretical research and education into condensed matter systems with strong correlations, utilizing mathematically sophisticated methods. A main theme is to find and understand systems with topological order. Topological order exists in two-dimensional electron gases. A closely related phase occurs in three-dimensional topological insulators. Another promising possibility for realizing topological order is in spin liquids. The PI aims to derive exact results for quasiparticles with non-Abelian statistics, where the physics depends on the order in which the quasiparticles are interchanged. The PI also seeks to find lattice models with topological order by exploiting the close connection to two-dimensional classical integrable models. Another theme is to develop new mathematical tools to study topologically ordered and other new states of matter. The PI intends to exploit symmetries that will make exact computations possible. One promising approach is to study models with both integrability and supersymmetry; these models allow the use of mathematical results from disparate areas of mathematics such as combinatorics, topology, and non-linear differential equations. This research lies at an interface of mathematics and condensed matter physics. Predicted and discovered phenomena and new states of matter increasingly require more sophisticated mathematics for their description. The PI will continue to collaborate with mathematicians; the result may lead to advances in condensed matter physics and to new ideas in mathematics, particularly in topology, combinatorics and analysis. NON-TECHNICAL SUMMARY The Division of Mathematical Sciences and the Division of Materials Research contribute funds to this award. It supports research and education in exciting current problems in condensed-matter theory which are leading to discoveries of new states of matter. The PI will focus on new states of matter that extend our understanding of what constitutes distinguishable phases of matter. The field of topology provides mathematical ideas that together with quantum mechanics lead to deeper understanding of some new states of matter. These include recently discovered robust metallic states that appear on the surface of a class of insulating materials, eventhough the bulk of the material remains an insulator. The metallic states of these "topological insulators" have distinctive properties; among them is the ability to carry electric currents without dissipation. The PI strives for a deeper understanding of the nature of another state of matter that arises in electrons confined to two dimensions in artificial semiconductor materials and placed in a high magnetic field. Under appropriate conditions, a new state of matter is predicted to emerge from the strongly interacting electrons. Amazingly, this state appears to consist of charged particles, each one having a charge equal to a fraction of an electron charge. It has been proposed that manipulation of these fractionally charged "particles" confined to two dimensions can enable a new kind of computer technology that is based on quantum mechanics and is many times faster than existing computers for some important kinds of problems. This research lies at an interface of mathematics and condensed matter physics. Predicted and discovered phenomena and new states of matter increasingly require more sophisticated mathematics for their description. The PI will continue to collaborate with mathematicians; the result may lead to advances in condensed matter physics and to new ideas in mathematics, particularly in topology, combinatorics and analysis.