Coulomb Gases and Vortex Systems: Two-Dimensional Physics and Beyond

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The Coulomb electrostatic interaction is one of the fundamental forces in nature, governing how elementary charged particles interact. For example, stars are essentially plasmas - that is gases of positively charged ions and negatively charged electrons attracting and repelling one another according to the Coulomb force. The formation of crystals, which are periodic arrangements of atoms, can be roughly explained via repulsive electrostatic (Coulomb) forces coupled with a binding force. In Physics, a "two-component plasma" is a large collection of positively charged particles and as many negatively charged particles; a "one-component plasma" is a large collection of positively charged particles with the negative particles replaced by a uniform neutralizing background. Such systems are also called "Coulomb gases," and they are quite important in theoretical physics. Physicists predicted that in two-dimensional Coulomb gases, a new intermediate state of matter - formed of dipoles of oppositely charged particles - should exist between the totally ordered (solid) and disordered (liquid) state, once a certain critical temperature is reached. This was a new type of phase transition in statistical physics. It is specific to dimension 2 and an important example of two-dimensional physics. This project is particularly concerned with the analysis of large Coulomb gases in dimension 2 and higher, both one-component and two-component, in the framework of statistical mechanics (that is, with temperature), and using the techniques of mathematical analysis and probability theory. Also motivating such questions are the study of energy levels of large atoms (spectrum of large random matrices), vortices in superconductors and Bose-Einstein condensates, the fractional quantum Hall effect, and more loosely questions in biology or hydrodynamics. The project also contains a component on the mathematical analysis of vortices in superconductors in the context of the famous Ginzburg-Landau model, another and related instance of two-dimensional physics, with the investigation of this phenomenon in dimension 3. This project provides research training opportunities for graduate students. This project is about the mathematical analysis of large systems of points, or lines, with Coulomb interactions. Such systems are ubiquitous in physics models, ranging from quantum mechanics, statistical mechanics to plasma physics and condensed matter physics, but also in random matrix theory and approximation theory. The main part of the proposal concerns the statistical mechanics of classical Coulomb gases. In a first part it deals with positively charged points. The main questions are to investigate the effect of temperature on the local structure of the point configurations, in broad temperature regimes, as well as to understand their possibly universal features. While a lot has been understood in dimension 2, much remains to be done to understand which properties persist in higher dimension, in particular to understand the order of the charge fluctuations and their possibly Gaussian nature, as well as the long-range correlations of the system and free energy expansion. The proposal also aims at understanding how much of the features are specific to the Coulomb interaction by investigating the same questions for Riesz gases. A second part concerns the two-dimensional Coulomb gas with opposite charges (or two-component plasma). This is a very important model, related to the XY model in which a similar transition happens, itself an approximation for two-dimensional systems in condensed matter physics including Josephson junction arrays and thin disordered superconducting granular films. This BKT transition is charcterized by the emergence of dipoles and the transition from exponentially decaying to algebraically decaying correlations. A lot of this behavior is awaiting further mathematical proofs and this is what the project will try to address in the context of the two-component Coulomb gas. Finally the last part of the proposal concerns lines with Coulomb interactions, which arise as vortices in three-dimensional models from superconductivity and superfluidity. The geometry of the lines makes the analysis much more delicate than for points. While many mathematical tools originating in geometric measure theory have been developed for dealing with that aspect, the Coulomb interaction of the lines, the characterization of the onset of vortex lines under an applied magnetic field, and their collective behavior still need to be further investigated. The project's goal is to study all these questions with rigorous mathematical proofs and tools that will mix analysis, PDE, calculus of variations, and probability.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.