  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.
