  A model of $N$ 4-component massless fermions in a quartic self-interaction
based on ref. \cite{gomes2022} is investigated in the presence of chemical
potential and temperature via optimized perturbation theory that accesses
finite-N contributions. We use the generating functional approach to calculate
the corrections to the effective potential of the model. The model introduces
an auxiliary pseudo-vector field with a nontrivial minimum and is influenced by
temperature $(T)$ and chemical potential $(\mu)$. These thermodynamic
quantities are introduced through Matsubara formalism. Thereby, the integrals
are modified, and via the principle of minimum sensitivity, we obtain the gap
equations of the model. The correspondent finite-N solutions of these equations
define the vacuum states of the model associated with the background
pseudo-vector field. In particular, one focuses on its temporal component that
acts as an effective chiral chemical potential. We discuss the solutions of the
four cases in which $(T = 0,\mu = 0)$, $(T \neq 0,\mu \neq 0)$, $(T \neq 0,\mu
= 0)$ and $(T = 0,\mu \neq 0)$, where the effective potential is so obtained as
a function of the background vector field, the chemical potential, and the
temperature. The model shows the finite-N corrections generate first-order
phase transitions on the self-interacting fermions for the case $N=1$ and the
persistence of a second-order phase transition for $N \geq 2$.
