Exact Solution of Strongly Interacting Quasi-One-Dimensional Spinor Bose Gases

We present an exact analytical solution of the fundamental system of quasi-one-dimensional spin-1 bosons with infinite $\delta$ repulsion. The eigenfunctions are constructed from the wave functions of noninteracting spinless fermions, based on Girardeau’s Fermi-Bose mapping. We show that the spinor bosons behave like a compound of noninteracting spinless fermions and noninteracting distinguishable spins. This duality is especially reflected in the spin densities and the energy spectrum. We find that the momentum distribution of the eigenstates depends on the symmetry of the spin function. Furthermore, we discuss the splitting of the ground state multiplet in the regime of large but finite repulsion.

Figure 1

Cut through the nonzero spin components of a bosonic 3-particle state (see text). The second and the third coordinate are fixed at $z_2=-0.8l$ and $z_3=0.8l$. Shown is the exact wave function in the limit of infinite $\delta$ repulsion (red solid line) and the solution of a numerical diagonalizion of (1) in the limit of large but finite repulsion (blue dashed line).

Figure 2

Spin densities of 8 spin-1 bosons in different ground states (see text). Shown are the densities ${\rho }^{(i)}$ (gray dash-dotted line, see text), and the components ${\rho }_0$ (red solid line), ${\rho }_1$ (blue dashed line), and ${\rho }_{-1}$ (green dotted line) of the spin density. The spin densities resemble the one of a chain of localized spins.

Figure 3

Momentum distributions of 5 spin-1 bosons in different ground states. The gray dashed line shows for comparison the momentum distribution of 5 noninteracting fermions. The shape of the momentum distribution depends on the symmetry of the spin function.