Correlations of the metastable branch of harmonically-trapped one-dimensional two-component Fermi gases

We present highly-accurate energy spectra and eigen functions of small harmonically-trapped one-dimensional two-component Fermi gases with interspecies $\delta$-function interactions, and analyze the correlations of the metastable crossover branch with effectively repulsive and attractive interactions. Structural expectation values are presented as a function of the interspecies coupling strength and related to cold atom tunneling experiments. Moreover, we show that the eigenstate $\psi_{adia}$ 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, and $n_1+n_2 = n$), which is reached experimentally by adiabatically changing the system parameters, does not, as previously proposed, coincide with the wave function $\psi_{G}$ obtained by applying a generalized Fermi-Fermi mapping function to the eigen function of the non-interacting single-component Fermi gas.