Fermi gas with attractive potential and arbitrary spin in a one-dimensional trap: Phase diagram for S=3/2, 5/2, 7/2, and 9/2

A gas of ultracold Li-6 atoms (effective spin 1/2) confined to an elongated trap with one-dimensional properties is a candidate to display three different phases: (i) fermions bound in Cooper-pair-like states, (ii) unbound spin-polarized particles, and (iii) a mixed phase in which Cooper bound states and unpaired particles coexist. It is of great interest to extend these studies to fermionic atoms with higher spin, e. g., for neutral K-40, Ca-43, Sr-87, or Yb-173 atoms. Within the grand-canonical ensemble, we investigated the mu versus H phase diagram (mu is the chemical potential and H the external magnetic field) for S = 3/2, ... , 9/2 for the ground state using the exact Bethe ansatz solution of the one-dimensional Fermi gas with an attractive delta-function interaction potential. There are N = 2S + 1 fundamental states: the particles can be either unpaired or clustered in bound states of 2, 3,..., 2S, and 2S + 1 fermions. The rich phase diagram consists of these N states and various mixed phases in which combinations of the fundamental states coexist. Bound states of N fermions are not favorable in high magnetic fields, but always present if the field is low. For S = 3/2, possible scenarios for phase separation are explored within the local density approximation. For S = 3/2, the phase diagram for the superposition of a Zeeman and a quadrupolar splitting is also discussed.