Three attractively interacting fermions in a harmonic trap: Exact solution, ferromagnetism, and high-temperature thermodynamics

Three fermions with strongly repulsive interactions in a spherical harmonic trap constitute the simplest nontrivial system that can exhibit the onset of itinerant ferromagnetism. Here, we present exact solutions for three trapped, attractively interacting fermions near a Feshbach resonance. We analyze energy levels on the upper branch of the resonance where the atomic interaction is effectively repulsive. When the s-wave scattering length a is sufficiently positive, three fully polarized fermions are energetically stable against a single spin-flip, indicating the possibility of itinerant ferromagnetism, as inferred in the recent experiment. We also investigate the high-temperature thermodynamics of a strongly repulsive or attractive Fermi gas using a quantum virial expansion. The second and third virial coefficients are calculated. The resulting equations of state can be tested in future quantitative experimental measurements at high temperatures and can provide a useful benchmark for quantum Monte Carlo simulations.