Unitary quantum gases: from cold atoms to quark-gluon plasmas

We investigate the many-body properties of two distinct degenerate systems with strong interactions, namely that of a quark-gluon plasma and of an atomic Bose gas. In the first part of this thesis, the temperature dependence of the thermodynamic potential of quantum chromodynamics is studied. In particular, the specific heat and the quark effective mass are calculated for imbalanced quark matter in the limit of a large number of quark flavors (large-Nf), which corresponds to the random phase approximation. Also, a generalization of the Landau effective-mass relation for the imbalanced case within relativistic Landau-Fermi liquid theory is derived. The effects of imbalance is expected to have important consequences on the properties of quark matter in the core of neutron stars. In the second part, we develop an analytical approach for the description of the crossover of an atomic Bose gas from small to infinitely large scattering length (unitarity). We obtain several properties of the Bose gas as a function of interaction strength, such as the chemical potential, the contact, the speed of sound, the condensate density, the effective interatomic interaction and the three-body recombination rate. It is shown how the approach can be systematically improved with renormalization-group methods and that it reduces to the Bogoliubov theory in the weak-coupling limit.