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We study relativistic fermionic systems in $3+1$ spacetime dimensions atfinite chemical potential and zero temperature, from a path-integral point ofview. We show how to properly account for the $i\varepsilon$ term that projectson the finite density ground state, and compute the path integral analyticallyfor free fermions in homogeneous external backgrounds, using complex analysistechniques. As an application, we show that the ${\rm U}(1)$ symmetry is alwayslinearly realized for free fermions at finite charge density, differently fromscalars. We study various aspects of finite density QED in a homogeneousmagnetic background. We compute the free energy density, non-perturbatively inthe electromagnetic coupling and the external magnetic field, obtaining thefinite density generalization of classic results of Euler--Heisenberg andSchwinger. We also obtain analytically the magnetic susceptibility of arelativistic Fermi gas at finite density, reproducing the de Haas--van Alpheneffect. Finally, we consider a (generalized) Gross--Neveu model for $N$interacting fermions at finite density. We compute its non-perturbativeeffective potential in the large-$N$ limit, and discuss the fate of the ${\rmU}(1)$ vector and $\mathbb{Z}_2^A$ axial symmetries.

Fermions at finite density in the path integral approach