Improving the grasp of harmonically trapped fermions in low dimensions

Experiments with only a few atoms shed light on the crossover from few- to many-body physics. Theoretical treatment of these experiments is challenging because the underlying Hilbert space grows rapidly with particle number. The standard effective treatment for short-ranged interactions relies on the δ-potential, for which the exact diagonalization in a truncated Hilbert space converges slowly. Moreover, in spatial dimensions larger than one, the δ-potential requires careful renormalization to converge at all. Here we exploit a particularly effective renormalization procedure adapted from nuclear physics: We fix the lowest part of the two-body spectrum using the exact solution. This provides us with an efficient tool to obtain physical observables free from a cutoff dependence at significantly reduced computational cost.

To demonstrate our approach, we show results for a few harmonically trapped fermions in 1D interacting with a static magnetic impurity at the center of the trap. By gradually increasing the particle number in the trap, a quantum phase transition is approached, manifesting itself in a level crossing between two ground states with vanishing spin and non-zero spin, respectively. Moreover, we address low-energy physics of harmonically trapped 2D Fermi gases allowing us to study the nature of particle pairing in this regime.