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.
Our most complete understanding of the universe in which we live is summarized in the "standard model" of particle physics. In this model the fundamental bits of matter – quarks and leptons – interact through gauge fields which are generalizations of electricity and magnetism. Some of these interactions (the "weak" force) possess the curious feature that they only affect the left-handed parts of the matter. A mathematically natural way for this handedness to arise is for matter (indeed our entire universe) to be trapped on a surface embedded in some higher dimensional space. This notion sounds as if it belongs in a science fiction story, but in recent years quite down-to-earth researchers have realized that certain crystalline insulators containing heavy elements behave in exactly this way. These "topological insulators" are only insulators on their inside. On their surfaces, electrons are free to move and conduct electricity, but they do so in a manner that forms a precise analogue of the way that quarks interact with the weak gauge field. These electrically conducting surface states are of great interest, partly for practical reasons in building new generations of solid-state electronic devices that might lead to quantum computers, but also because the analogy with particle physics allows us to explore speculative particle physics at a much lower cost than building large accelerators. The work funded by this proposal investigated several aspects of these topological materials.. Firstly we explored the mathematics behind the bulk-surface connection. This mathematics, topological K-theory, is much more sophisticated than that familiar to most solid state physicists, and this is why no-one noticed the possibility of these materials in the 75 years between the initial work on electrically conducting crystals in the late 1920's and the discovery of topological insulators in the mid-2000's. We were able to explain much of the mathematics of K theory in simple terms, and as a consequence we are able to extend previously known results to investigate the surface states of new symmetry classes of crystals. Secondly we sought to use what we know already of these topological materials to explore their particle and gravitational physics analogies. We explored connections between heat conduction by the surface states and the thermal properties of black holes. One outcome of this investigation was our proposal of a novel laborarory-scale anaogue of a black hole. Just as Stephen Hawkin suggested in the 1970's the event horizon of our model black hole emits thermal radiation. The temperature of this radiation is many orders of magnitude higher that that expected from other analogue models based on Bose condensed atomic gasses. We also investigated how the dynamics of the spin of the surface electrons affects their motion, We were able to provide a novel derivation of the gauge "anomaly" that leads to the exchange of particles between the surface states and bulk of the crystal. In particle physics the necessity of the cancellation of these gauge anomalies provide important consistency constraints in the standard model. A secondary project, rather distinct from the above program but still in the realm of topologically interesting materials , was to provide a detailed numerical solution of the interaction between superconductivity and electromagentism for systems with "p-wave'' superconductivity. The reason for this work was to provide support for the experimental claim by our colleague Raffi Budakian for the existence of "half quantum" vortices in superconducting strontium ruthenate. We were able to use the equations for a p-wave superconductor to qualitatively reproduce the experimental data. This provides good evidence that strontium ruthenate is a p-weve superconductor, and therefore a candidate topological solid.
