The many-body problems is at the heart of a great variety of  macroscopic condensed matter phenomena.  Our activity aims at the description of generic, classical and quantum,  N-body systems, ranging from  electrons in solids, cold atomic gases, and quantum magnetism  to the microscopic and coarse grained description of  equilibrium and non-equilibrium phase transitions.

Conceptually, the easiest description is often based on mean-field approaches.  Their rigorous derivations  and extensions in the context of physics of cold atomic gases  is part of our activity. The main impetus is, however, to develop and apply beyond mean field methods.

These methods serve  to include quantum and thermal fluctuations, to establish and quantify possible  phase diagrams of Bose and Fermi gases and their mixtures from the dilute to the strongly interacting  regimes. Our numerical solutions allow us direct,  parameter-free  comparisons with experimental results. (link to atomic gases).  We further study the possibility of Cooper pair and quartett condensation in fermionic gases  in analogy to phenomena in  strongly interacting nuclear matter. Indeed, if  there are systems with four species of fermions, two Cooper pairs can form a quartet. Competition between pairing and quartetting is then an interesting phenomenon to  be investigated.

Advanced many-body approaches are also developed to describe  strong correlations in quantum magnetism and spin systems. 

Renormalization group theory has been particularly successful for the explanation of  phase transitions. We mainly develop non-perturbative, functional renormalization  group  methods, extending them from equilibrium to non-equilibrium critical  phenomena, such of kinetic roughening or absorbing transitions in reaction-diffusion processes (link to theme hors equilibre). 

Another activity at LPMMC concerns  the theoretical investigation of quantum  many-body Hamiltonians. We study some of their fundamental mathematical  properties with applications e.g. to the modelization of quantum crystals and  polarons.