Correlations of the upper branch of 1d harmonically trapped two-component Fermi gases

We present highly-accurate energy spectra and eigen functions of small 1d harmonically trapped two-component Fermi gases with interspecies $\delta$-function interactions, and analyze the correlations of the so-called upper branch (i.e., the branch that describes a repulsive Fermi gas consisting of atoms but no molecules) for positive and negative coupling constants. Changes of the two-body correlations as a function of the interspecies coupling strength reflect the competition of the interspecies interaction and the effective repulsion due to the Pauli exclusion principle, and are interpreted as a few-body analog of a transition from a non-magnetic to a magnetic phase. Moreover, we show that the eigenstate $\psi_{\rm{adia}}$ of the infinitely strongly-interacting system with $|n_1+n_2|>2$ and $|n_1-n_2| < n$ ($n_1$ and $n_2$ denote the number of fermions of components 1 and 2, respectively), which is reached experimentally by adiabatically changing the system parameters, does not, as previously proposed, coincide with the wave function $\psi_{\rm{G}}$ obtained by applying a generalized Fermi-Fermi mapping function to the eigen function of the non-interacting single-component Fermi gas.