Theoretical Studies of Quantum Systems with Strong Interations

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TECHNICAL SUMMARYThis award is funded by the Division of Materials Research and the Physics Division. It supports theoretical research and education focused on geometrical non-equilibrium phenomena in coherent quantum liquids, especially in Fractional QuantumHall edge states; non-Abelian interference phenomena in quantum impurities, and geometrical analysis of singularities and singular patterns arising in Hamiltonian driven non-equilibrium processes such as Laplacian Growth and diffusion limited aggregation models.The research topics are unified by the goal to develop a theory of non-equilibrium and interference processes with underlying spatial conformal symmetry. This project is concerned with geometric analyses of singularities arising in non-equilibrium processes using advances achieved under prior NSF support. Many important systems out of equilibrium show conformal symmetries and therefore integrable structures similar to conformal invariance of critical phenomena. However non-equilibrium processes are different. Conformal invariance inevitably leads to singular patterns occurring at small scales. In its turn singularities give rise to fractal non-equilibrium patterns visible at large scales. The origin, statistics, and regularization of singularities and fractal geometry of stochastic patterns of driven processes comprise one theme of the research. Another theme is non-linear quantum hydrodynamics of Fractional Quantum Hall edge states. The emphasis is given to a topological manifestation of a fractional charge excitation as an edge soliton. The last theme of the research is non-Abelian interference phenomena as realized in controlled artificially fabricated quantum nanodevices exhibiting over-screened multichannel Kondo regime.The PI will integrate education and research through training and mentoring graduate and undergraduate research students, and making novel contributions to the Research Experiences for Undergraduates.NON-TECHNICAL SUMMARYThis award is funded by the Division of Materials Research and the Physics Division. It supports theoretical condensed matter physics research and education at an interface with mathematical physics and mathematics. The research is focused on advancing our understanding of complex non-equilibrium processes. An important aspect of the PI?s work involves growth processes that display snowflake-like fingers that penetrate from one phase into another, as happens in the growth of alloys and semiconductor structures. The PI seeks a fundamental understanding of how these fingering patterns emerge in the growth process. Capitalizing on subtle connections between seemingly disparate areas of research, the PI will also study new states of matter that emerge at the edges of a droplet of electrons in a high magnetic field. The PI will also pursue an approach to observing unusual new states of matter, topological states, in special tiny structures of atoms called quantum dots. These states were theoretically predicted to exist in electronic liquids confined to two dimensions and in high magnetic field; the PI?s proposal provides a new arena in which to study these possible new states of matter that may enable us to exploit quantum mechanical states to perform computation. Quantum computing is believed to provide an opportunity for a vast improvement in computer performance, at least on some important problems, cryptography being one example. The PI will integrate education and research through training and mentoring graduate and undergraduate research students, and making novel contributions to the Research Experiences for Undergraduates. The results of the proposed research will enhance knowledge and understanding of complex condensed matter systems that are far from being in an equilibrium state.