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Ultracold spinor atoms in the weak and strong interaction regime havereceived extensive investigations, while the behavior in the intermediateregime is less understood. We numerically investigate ultracold spinor atomicensembles of finite size in the intermediate interaction regime, and reveal theevolution of the eigenstates from the strong to the intermediate regime. In thestrong interaction regime, it has been well known that the low-lyingeigenenergy spectrum presents the well-gaped multi-manifold structure, and theenergy gaps protect the categorization of the eigenstates. In the intermediateinteraction regime, it is found that the categorization of the eigenstates ispreserved, and the eigenenergy spectrum become quasi-continuum, with differentmanifolds becoming overlapped. The overlapping induces both direct and avoidedcrossings between close-lying manifolds, which is determined by the combinedsymmetries of the eigenstates involved in the crossing. A modified t-J model isderived to describe the low-lying eigenstates in the intermediate regime, whichcan capture the formation and crossings of the manifolds. State preparationthrough the avoided crossings is also investigated.

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