Phase Structure of Strongly Correlated Fermi Gases

Strongly correlated fermionic many-body systems are ubiquitous in nature. Their theoretical description poses challenging problems which are further complicated when imbalances in, e.g., the particle numbers of the involved species or their masses are introduced. In this thesis, a number of different approaches is developed and applied in order to obtain predictions for physical observables of such systems that mutually support and confirm each other. In a first step, analytically well-founded mean-field analyses are carried through. One- and three-dimensional ultracold Fermi gases with spin and mass imbalance as well as Gross-Neveu and NJL-type relativistic models at finite baryon chemical potential are investigated with respect to their analytic properties in general and the occurrence of spontaneous breaking of translational invariance in particular. Based on these studies, further methods are devised or adapted allowing for investigations also beyond the mean-field approximation. Lattice Monte Carlo simulations with imaginary imbalance parameters are employed to surmount the infamous sign problem and compute the equation of state of the respective unitary Fermi gases. Moreover, in-medium two-body analyses are used to confirm and explain the characteristics of inhomogeneously ordered phases. Finally, functional RG methods are applied to the unitary Fermi gas with spin and mass imbalance. Besides quantitatively competitive predictions for critical temperatures for the superfluid state, strong hints on the stability of inhomogeneous phases with respect to order parameter fluctuations in the regime of large mass imbalance are obtained. Combining the findings from these different theoretical studies suggests the possibility to find such phases in experiments presently in preparation