Constructions of Fermionic Field Theories
Fermionic field theories are ubiquitous in condensed matter theory and low-temperature physics. For example, the electronic structure of metals and semiconductors falls into the class of correlated electron systems, as do cold atomic gases made from atoms with half-integer spin.
Starting with pioneering work by J. Feldman and E. Trubowitz, the methods of constructive quantum field theory and the mathematical renormalization group have been used to prove physical properties of such models, in the parameter regions where these models are also interesting in applications. An important advantage of these methods is that they give control over the correlation functions of the system, also at positive temperatures. Most prominently, Landau's theory of Fermi liquids has been justified in a class of models, and symmetry breaking is being studied in the context of magnets and superconductors.
Our research has contributed significantly to the regularity analysis of the fermionic self-energy and the Fermi surface and to the mathematically rigorous identification of Fermi liquid behaviour. We have also contributed to simplifying and extending the methods used (see, e.g., W. Pedra and M. Salmhofer, Commun. Math. Phys. 282 (2008) 797). Current projects include the study of diilute fermion systems (to appear), a general proof of Fermi liquid behaviour for systems with strictly convex Fermi surface in two dimensions (W. Pedra, PhD thesis), and the extension of these methods from equilibrium correlation functions to response functions.