This award supports theoretical research and education with an aim to develop nonperturbative methods for systems with strong interactions. The PI will focus on the hydrodynamic approach which provides a general framework for treating interacting systems. The PI aims to develop such an approach to gain insight into quantum low-dimensional strongly interacting systems focusing on integrable models. The linearized version of the hydrodynamic description, bosonization, is very effective in one-dimension. The PI plans to extend this approach to nonlinear and dispersive hydrodynamic theory. In contrast to linear bosonization, the nonlinear theory will be suitable for studying nonlinear effects, such as formation of dispersive shock waves. Shock waves have been observed in systems of cold atoms, and these new tools would be timely. The PI will study several physical systems such as cold atoms, quantum dots, two-dimensional electron gas in quantum Hall regime, and spin chains using topological methods and exact results for quantum mesoscopic transport.
The PI is writing a review on the use of topological methods in quantum condensed matter physics and delivers tutorial lectures to researchers in condensed matter physics. The PI is developing a new course on topological aspects in solid state physics which will be taught to physics graduate students in Stony Brook. The PI teaches mathematics in K-12 enrichment program and teaches physics and mathematics to gifted high school students in Russia.
NONTECHNICAL SUMMARY This award supports theoretical research and education with the aim of developing a new approach for strongly interacting quantum mechanical systems, like electrons in strongly correlated materials and low temperature gases of atoms trapped by laser beams. The PI's approach builds on a method that has been successfully applied to systems confined to one dimension. The PI will use these methods to study some of the most challenging problems in condensed matter physics, such as spin chains and the nature of new states of matter predicted to exist in gases of electrons confined to dimensions in semiconductor crystals in a perpendicular strong magnetic field. These topological states of matter may enable a new kind of computation based on the manipulation of quantum mechanical states. Unlike other proposals for quantum computing, topological states would be comparatively immune from environmental effects that would interfere with the operation of a quantum computer.
The PI is writing a review on the use of topological methods in quantum condensed matter physics and delivers tutorial lectures to researchers in condensed matter physics. The PI is developing a new course on topological aspects in solid state physics which will be taught to physics graduate students in Stony Brook. The PI teaches mathematics in K-12 enrichment program and teaches physics and mathematics to gifted high school students in Russia.
At very low temperatures ordinary atoms behave in a very strange way dictated by the laws of quantum mechanics. In 2001 the Nobel prize in physics was awarded to E. Cornell, W. Ketterle, and C. Wieman for observing the new state of matter – Bose Einstein condensation – in the system of ultracold atoms. Since then there was a flurry of beautiful experiments studying the properties of ultracold atomic systems. In one of those experiments J. Thomas and J. Joseph observed an unusual behavior during the collision of two clouds of very cold lithium atoms. The atoms were suspended in space using magnetic fields and lasers. Two atomic clouds have been released and collided with the time evolution recorded in the form of cloud density images. The picture shows the result of such collisions. We assumed that in the limit of extreme interactions the cold atom gas behaves as a fluid. Simulating the behavior of the gas by one-dimensional hydrodynamic equations we identified the sharp boundaries between regions with high and low densities of atoms clearly seen on the picture as shock waves. The agreement of theoretical simulations with experiment is very good which proves that (i) the gas can be described by fluid (ii) the identification of the sharp boundaries with shock waves is correct. We also found a hydrodynamic model describing Fractional Quantum Hall effect as a dynamics of a fluid made of electrons. The effect occurs when electrons confined to the surface of semiconductor are subject to a strong magnetic field perpendicular to the surface. It was known for long time that such electrons behave in ways similar to the incompressible fluid. We extended such a description and constructed the model which shows Hall viscosity. Hall viscosity is believed to be present in Hall effect. It describes the appearance of the pressure forces perpendicular to the velocity of the fluid which occurs in the presence of shear, i.e., when layers of the fluid are being displaced relative to each other. We also studied the statistics of a charge transport in quantum mechanical systems. We developed the formalism of computing the full statistic of such transport in the simplest systems consisting of fermionic particles (having the same quantum statistics as electrons) confined to one-dimensional constrictions. P.I. has continued to actively engage in outreach and education. In July-August 2012 he taught mathematics and physics in Krasnoyarsk Summer School for gifted high school students (Russia, Siberia). He gave lectures to students introducing noneuclidian geometry and gave a course "Knots and Braids" involving lectures and hand-on sessions on mathematics and practical applications of knots.
