  We calculate the finite-temperature density and polarization equations of
state of one-dimensional fermions with a zero-range interaction, considering
both attractive and repulsive regimes. In the path-integral formulation of the
grand-canonical ensemble, a finite chemical potential asymmetry makes these
systems intractable for standard Monte Carlo approaches due to the sign
problem. Although the latter can be removed in one spatial dimension, we
consider the one-dimensional situation in the present work to provide an
efficient test for studies of the higher-dimensional counterparts. To overcome
the sign problem, we use the complex Langevin approach, which we compare here
with other approaches: imaginary-polarization studies, third-order perturbation
theory, and the third-order virial expansion. We find very good qualitative and
quantitative agreement across all methods in the regimes studied, which
supports their validity.
