Fermions at finite density in the path integral approach

We study relativistic fermionic systems in 3 + 1 spacetime dimensions at finite chemical potential and zero temperature, from a path-integral point of view. We show how to properly account for the iε term that projects on the finite density ground state, and compute the path integral analytically for free fermions in homogeneous external backgrounds, using complex analysis techniques. As an application, we show that the U(1) symmetry is always linearly realized for free fermions at finite charge density, differently from scalars. We study various aspects of finite density QED in a homogeneous magnetic background. We compute the free energy density, non-perturbatively in the electromagnetic coupling and the external magnetic field, obtaining the finite density generalization of classic results of Euler-Heisenberg and Schwinger. We also obtain analytically the magnetic susceptibility of a relativistic Fermi gas at finite density, reproducing the de Haas-van Alphen effect. Finally, we consider a (generalized) Gross-Neveu model for N interacting fermions at finite density. We compute its non-perturbative effective potential in the large-N limit, and discuss the fate of the U(1) vector and ${\mathbb{Z}}_2^A$ \( {\mathbb{Z}}_2^A \) axial symmetries.