Correlations of the Upper Branch of 1D Harmonically Trapped Two-Component Fermi Gases

We present highly accurate energy spectra and eigenfunctions 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 nonmagnetic to a magnetic phase. Moreover, we show that the eigenstate ${\psi }_{\mathrm{adia}}$ 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 ${\psi }_G$ obtained by applying a generalized Fermi-Fermi mapping function to the eigenfunction of the noninteracting single-component Fermi gas.

Figure 1

Relative energies for the (a) (2, 1) system with ${\Pi }^{\mathrm{rel}}=-1$, the (b) (3, 1) system with ${\Pi }^{\mathrm{rel}}=-1$, and the (c) (2, 2) system with ${\Pi }^{\mathrm{rel}}=+1$ as a function of $-1/g$. The dashed lines show the eigenenergies corresponding to states that are not affected by the interspecies interactions. The thick solid lines show the upper branch.

Figure 2

Relative wave function of the (2, 1) system with $1/|g|=0$ and ${\Pi }^{\mathrm{rel}}=-1$. Contour plots of (a) ${\psi }_{\mathrm{adia},1}^{|g|=\infty }$ and (b) ${\psi }_{G,0}$ as functions of $z_{12}$ and $z_{12,3}$. Nodal lines are shown by solid lines. The dashed and dotted contours indicate positive and negative wave function regions; the contours are spaced equidistantly. (c) Dotted and solid lines show cuts of ${\psi }_{\mathrm{adia},1}^{|g|=\infty }$ and ${\psi }_{G,0}$ as a function of $z_{12}$ for $z_{12,3}=a_{\mathrm{ho}}$. The thin dashed vertical lines at $z_{12}=\pm \sqrt{3}{a}_{\mathrm{ho}}$ are shown as a guide to the eye.

Figure 3

Expectation values $⟨|z_{jk}|⟩$ for the up-up (solid lines) and up-down distance coordinate (dotted lines) as a function of $-1/g$ for the upper branches of the (a) (2,1), (b) (3, 1), and (c) (2, 2) systems.