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

TL;DR

This paper explores the phase diagram of a one-dimensional attractive Fermi gas with arbitrary spin, revealing complex phases including bound states of multiple fermions, using exact Bethe ansatz solutions.

Contribution

It extends the analysis of one-dimensional Fermi gases to higher spins, providing a detailed phase diagram with bound states and mixed phases using exact solutions.

Findings

Four fundamental states identified: unpaired, two-, three-, and four-fermion bound states.

Bound states of four fermions are suppressed at high magnetic fields.

Universal phase diagram obtained through scaling and local density approximation.

Abstract

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.