Fermi gas with attractive potential and arbitrary spin in one-dimensional trap

A gas of ultracold $^6$Li 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 which is believed to have some resemblance to the FFLO pairing. It is of great interest to extend these studies to fermionic atoms with higher spin, e.g., for neutral $^{40}$K, $^{43}$Ca, $^{87}$Sr or $^{173}$Yb atoms. Within the grand-canonical ensemble we investigated the $\mu$ vs. $H$ phase diagram for $S=3/2$ ($\mu$ is the chemical potential and $H$ the external magnetic field) for the ground state using the exact Bethe {\it ansatz} solution of the one-dimensional Fermi gas interacting with an attractive $\delta$-function potential. There are four fundamental states: The particles can be either unpaired or clustered in bound states of two, three and four fermions. The rich phase diagram consists of these four states and various mixed phases in which combinations of the fundamental states coexist. Bound states of four fermions are not favorable in high magnetic fields, but always present if the field is low. Working within the grand-canonical ensemble has the following advantages: (1) A universal phase diagram is obtained by scaling with respect to the interaction strength and (2) possible scenarios for phase separation are explored within the local density approximation. The phase diagram for the superposition of a Zeeman and a quadrupolar splitting is also discussed.