Nonlinear effects in quantum condensed matter systems

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TECHNICAL SUMMARYThis award supports theoretical research and education on strongly correlated electron systems. To make any progress in understanding systems with strong interactions nonperturbative methods are needed. The hydrodynamic approach provides a general framework for treating interacting quantum and classical systems.Hydrodynamic methods and methods developed in the field of classical integrable equations will be used to analyze nonlinear behavior of quantum interacting systems. The PI aims to develop a hydrodynamic description of cold atoms, fractional quantum Hall effect systems, and low-dimensional integrable models. The linearized version of the hydrodynamic description - bosonization is known to be very effective in one- dimensional physics. The PI will develop a nonlinear and dispersive hydrodynamic theory to study nonlinear effects, such as formation of dispersive shock waves and to describe higher-dimensional systems. The PI will also develop topological methods in condensed matter physics and apply exact results for quantum transport and integrable systems to topological insulators and superconductors, and to mesoscopic systems.The research lies at the interface with mathematics and provides a good environment for training graduate students. The PI is developing a course on topological terms for advanced graduate students and researchers and plans to make corresponding lecture notes publicly available in the near future. The PI teaches math and physics with enthusiasm to K-12 students in the enrichment program for children at Stony Brook and to gifted high school students in Russia.NON-TECHNICAL SUMMARY This award supports theoretical investigations of quantum mechanical systems of many electrons or atoms that interact strongly with each other. This remains a challenging and fruitful area of research. The PI aims to develop a hydrodynamic description of these systems. Within the framework of hydrodynamics, systems of electrons or atoms are viewed as fluids with effective properties, for example, density, velocity, and viscosity. As a result, well known phenomena in fluids have analogs in systems of very cold atoms and electrons. Examples include vortices, shock waves, and turbulence. The PI aims to develop a hydrodynamic approach to systems of cold atoms, to the fractional quantum Hall effect and to the "fluids" that arise in exactly solvable models. The fractional quantum Hall effect arises in electrons confined to two dimensions in artificial semiconductor structures or in a single layer of carbon atoms called graphene, and exposed to a high magnetic field. Under these conditions the interactions between electrons can become very strong. Research on quantum Hall systems has led to the notion of a new kind of order known as topological order which is reflected in topological insulators. Like ordinary insulators, for example rubber, topological insulators do not conduct electricity though the interior of the material. Unlike ordinary insulators, topological insulators are able to conduct electricity on their edges or boundaries through the formation of a new state of matter. Among the known topological insulators are compounds made of the elements bismuth and selenium, and bismuth and tellurium. The PI will also develop new theoretical methods and methods known in exactly solvable models and apply them to topological insulators and other states of electronic matter. The research lies at the interface with mathematics and provides a good environment for training graduate students. The PI is developing a course on topological terms for advanced graduate students and researchers and plans to make corresponding lecture notes publicly available in the near future. The PI teaches math and physics with enthusiasm to K-12 students in the enrichment program for children at Stony Brook and to gifted high school students in Russia.