The Mathematics of Interacting Fermions

ongoing

The quantum many-body problem presents us with a baffling variety of phenomena whose mathematical understanding is just leaving infancy. One of the most prominent examples is the behavior of electrons in condensed matter: surprisingly, despite the presence of strong interactions between particles in the microscopic Schroedinger equation, on a macroscopic level one observes almost non-interacting particles. Moreover, some properties even turn out to be universal, i.e., do not depend on the details of the microscopic equation at all. Fermi liquid theory has been phenomenologically developed as an emergent theory to describe these correlation effects in systems of interacting fermionic particles. The first goal of this project is a rigorous derivation of Fermi liquid theory from the Schroedinger equation. My approach will be based on the analysis of high-density scaling limits. While the analysis of scaling limits has been tremendously successful in the last years for bosonic systems, in fermionic systems it has been restricted to the derivation of mean-field theories. Recently I have developed an approximate bosonization for three-dimensional systems which can be rigorously applied in high-density scaling limits. This is one of the few tools that permit an analysis beyond mean-field theory, enabling us now to describe correlations without relying on perturbation theory. The second goal is to show that one-dimensional systems can be analyzed similarly but display a very different behavior called Luttinger liquid, demonstrating that the approach allows to distinguish Fermi from non-Fermi liquids. Thus I will not only provide a unified justification of the non-interacting electron approximation in two and more dimensions, but also pave a new way to and partially resolve the classification problem of the fermionic phase diagram.