  In this work we analyze how effects of finite size may modify the
thermodynamics of a system of strongly interacting fermions that we model using
an effective field theory with four-point interactions at finite temperature
and density and look in detail at the case of a confining two-layer system. We
compute the thermodynamic potential in the large-$N$ and mean-field
approximations and adopt a zeta-function regularization scheme to regulate the
divergences. Explicit expansions are obtained in different regimes of
temperature and separation. The analytic structure of the potential is
carefully analyzed and relevant integral and series representations for the
various expressions involved are obtained. Several known results are obtained
as limiting case of general results. We numerically implement the formalism and
compute the thermodynamic potential, the critical temperature and the fermion
condensate showing that effects of finite size tend to shift the critical
points and the order of the transitions. The present discussion may be of some
relevance for the study of the Casimir effect between strongly coupled
fermionic materials with inter-layer interactions.
