  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 $i\varepsilon$ 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 ${\rm U}(1)$ 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 ${\rm
U}(1)$ vector and $\mathbb{Z}_2^A$ axial symmetries.
