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Phase diagram of harmonically confined one-dimensional fermions with attractive and repulsive interactions

We construct the complete $U$-$μ$ phase diagram for harmonically confined ultracold fermionic atoms with repulsive and attractive interactions. ($μ$ is the chemical potential and $U$ the interaction strength.) Our approach is based on density-functional theory, and employs analytical expressions for the kinetic and correlation energy functionals. For repulsive interactions our calculations confirm previous numerical studies, and complement these by closed expressions for all phase boundaries and characteristic lines of the phase diagram, and by providing an explanation and solution for difficulties encountered in earlier density-functional work on the same system. For attractive interactions we propose a new and accurate interpolation for the correlation energy, and use it to extend the phase diagram to $U<0$.