  We investigate a simple extension of the standard model (SM) in which the
dark matter consists of a feebly interacting fermion (FIMP), charged under a
new $Z_4$ symmetry, that is produced in the early Universe by the freeze-in
mechanism. The only other new particle included in the model is a singlet
scalar, also charged under the $Z_4$, which couples to the fermion via Yukawa
interactions and to the SM Higgs. The model is truly minimal, as it admits just
five free parameters: two masses and three dimensionless couplings. Depending
on their values, the freeze-in mechanism can be realized in different ways,
each characterized by its own production processes. For all of them, we
numerically study the relic density as a function of the free parameters of the
model and determine the regions consistent with the dark matter constraint. Our
results show that this scenario is viable over a wide range of couplings and
dark matter masses. This model, therefore, not only offers a novel solution to
the dark matter problem, but it also provides a minimal realization of
freeze-in for fermion dark matter.
