Introduction to the Renormalization Group with applications to non-relativistic quantum electron gases. (Lecturer: Prof. V. Rivasseau)

The problem of understanding the low temperatures properties of non-relativistic quantum electron gases is of fundamental importance in condensed matter theory. While the case of non-interacting electrons is perfectly understood since the early days of quantum mechanics, the effect of electron-electron interactions on the qualitative behavior of the system, even in cases where interactions are very weak, is still to be fully understood. Phenomena such as superconductivity and fractional quantum Hall effect, whose explanation requires the inclusion of electron-electron interaction effects, are still elusive and poorly understood. In the case of weak interactions, a natural approach to the problem is to expand the free energy and the correlation function as power series in the coupling constant, measuring the strength of the elctron-electron interaction. It is well know that such series suffers from infrared divergences, and its analysis requires subtle resummations, usually implemented via multiscale cluster expansion methods and renormalization group (RG). The approach to fermionic condensed matter systems based on constructive RG is an ongoing program to study systematically the properties of weakly interacting non-relativistic electrons at finite density in one, two and three dimensions. In one dimension interacting fermions have been proved to form a Luttinger liquid at zero temperature. In the two dimensional continuum, interacting electrons have been shown to be a Fermi liquid above the critical temperature, in the case a circular Fermi surface, and up to zero temperature, in the case of an asymmetric Fermi surface. On the 2D square lattice, interacting fermions above the critical temperature have been shown to display a transition from Fermi liquid to non-Fermi liquid behavior, as the density is increased from zero to one. In this course we will provide an introduction to fermionic constructive RG methods with applications to the study of the 2D fermionic jellium model in the presence of a rotationally symmetric ultraviolet cutoff, along the lines of M. Disertori and V. Rivasseau: Rigorous Proof of Fermi Liquid Behavior for Jellium Two-Dimensional Interacting Fermions, Phys. Rev. Lett. 85, 361 - 364 (2000). If time permits, extensions to the 2D Hubbard model at half filling will be discussed.