Abstract

The one-dimensional (1D) Fermi gas with repulsive short-range interactions provides an important model of strong correlations and is often amenable to exact methods. However, in the presence of confinement, no exact solution is known for an arbitrary number of strongly interacting fermions. Here, we propose a novel ansatz for generating the lowest-energy wavefunctions of the repulsive 1D Fermi gas in a harmonic potential near the Tonks-Girardeau (TG) limit of infinite interactions. We specialize to the case of a single impurity interacting with N majority particles, where we may derive analytic forms of the approximate wavefunctions. Comparing with exact numerics, we show that the overlap between the wavefunctions from our ansatz and the exact ones in the ground-state manifold exceeds 0.9997 for N up to 8. Moreover, the overlap for the ground-state wavefunction extrapolates to 0.9999 as N goes to infinity. Thus our ansatz is essentially indistinguishable from numerically exact results in both the few- and many-body limits. We derive an effective Heisenberg spin-chain model for the regime near the TG limit, within which our ansatz is exact. The energy of states in excited manifolds is calculated using a dynamical SO(2,1) symmetry, which provides an exact relation between states related by a scaling transformation. We finally show how our results for the wavefunctions and the energy spectrum can be detected in cold atomic gases via collective-mode, tunneling, and radio-frequency experiments.

Reference: J. Levinsen, P. Massignan, G. M. Bruun, and M. M. Parish, arXiv:1408.7096