Degeneracies in trapped two-component Fermi gases

We report on previously unobserved degeneracies in two-component equal-mass Fermi gases with zero-range interactions under isotropic harmonic confinement. Over the past 10 years or so, two-component Fermi gases with zero-range interactions have become a paradigm for modeling condensed matter systems, nuclear matter and neutron matter. We provide strong evidence that the eigen energies of the $(3,1)$ system consisting of three spin-up atoms and one spin-down atom are degenerate with the eigen energies of the $(2,2)$ system consisting of two spin-up atoms and two spin-down atoms for any $s$-wave scattering length $a_s$, including infinitely large, positive and negative $a_s$. Evidence for the existence of analogous degeneracies for larger systems is presented. The degeneracies evidenced in our study introduce a new class of highly-correlated systems for which such degeneracies, and thus an underlying symmetry, exist.