Manifold formation and crossings of ultracold lattice spinor atoms in the intermediate interaction regime

Ultracold spinor atoms in the weak and strong interaction regime have received extensive investigations, while the behavior in the intermediate regime is less understood. We numerically investigate ultracold spinor atomic ensembles of finite size in the intermediate interaction regime, and reveal the evolution of the eigenstates from the strong to the intermediate regime. In the strong interaction regime, it has been well known that the low-lying eigenenergy spectrum presents the well-gaped multi-manifold structure, and the energy gaps protect the categorization of the eigenstates. In the intermediate interaction regime, it is found that the categorization of the eigenstates is preserved, and the eigenenergy spectrum become quasi-continuum, with different manifolds becoming overlapped. The overlapping induces both direct and avoided crossings between close-lying manifolds, which is determined by the combined symmetries of the eigenstates involved in the crossing. A modified t-J model is derived to describe the low-lying eigenstates in the intermediate regime, which can capture the formation and crossings of the manifolds. State preparation through the avoided crossings is also investigated.