Exact Solution for Infinitely Strongly Interacting Fermi Gases in Tight Waveguides

We present an exact analytical solution of the fundamental systems of quasi-one-dimensional spin-$1/2$ fermions with infinite repulsion for an arbitrary confining potential. The eigenfunctions are constructed by the combination of Girardeau’s hard-core contacting boundary condition and group theoretical method, which guarantees the obtained states to be simultaneously the eigenstates of $S$ and $S_z$ and satisfy antisymmetry under odd permutation. We show that the total ground-state density profile behaves like the polarized noninteracting fermions, whereas the spin-dependent densities display different properties for different spin configurations. We also discuss the splitting of the ground states for large but finite repulsion.

Figure 1

The GS density distributions of Fermi gas in the limit of infinite repulsion with $N_{\uparrow }=3$ and $N_{\downarrow }=2$.

Figure 2

The GS momentum distributions $n_{\sigma }(k)$ for the system with $N_{\uparrow }=N_{\downarrow }=2$. Solid line corresponds to spin singlet, dotted line to ferromagnetic state and dashed line to the case of free fermion for comparison.

Figure 3

(a) Comparison of GS density distributions obtained by ED for the system with $N_{\uparrow }=2$ and $N_{\downarrow }=1$ with the analytical result of $S=1/2$ state. (b) The energy versus the interaction for different spin state. The density distribution for the ferromagnetic state (inset).