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We calculate the finite-temperature density and polarization equations ofstate of one-dimensional fermions with a zero-range interaction, consideringboth attractive and repulsive regimes. In the path-integral formulation of thegrand-canonical ensemble, a finite chemical potential asymmetry makes thesesystems intractable for standard Monte Carlo approaches due to the signproblem. Although the latter can be removed in one spatial dimension, weconsider the one-dimensional situation in the present work to provide anefficient test for studies of the higher-dimensional counterparts. To overcomethe sign problem, we use the complex Langevin approach, which we compare herewith other approaches: imaginary-polarization studies, third-order perturbationtheory, and the third-order virial expansion. We find very good qualitative andquantitative agreement across all methods in the regimes studied, whichsupports their validity.

Polarized fermions in one dimension: density and polarization from complex Langevin calculations, perturbation theory, and the virial expansion