Pair up or separate? Competition between attraction and spin-dependent external force in a Hubbard model

I numerically investigate the ground-state properties of a one-dimensional Fermi-Hubbard model with attractive interactions (U < 0), subject to a linear, spin-dependent external potential with a tunable slope. This potential creates oppositely located wells for spin-up and spin-down particles. I explore how the competition between attractive on-site interactions and the external potential leads to distinct ground-state phases with different spatial density profiles. As the slope increases, the system undergoes a sequence of transitions: from an unpolarized state, to a phase-separated configuration, and finally to a fully spin-separated phase. I identify the critical slope values that mark these transitions and show that they can be estimated analytically using the local-density approximation (LDA). These results reveal a rich interplay between interactions and external forces. I discuss potential realizations of this model in cold-atom systems and solid-state systems.