Abstract: 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 a 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 interlayer interactions.