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Effective spin-chain model for strongly interacting one-dimensional atomic gases with an arbitrary spin

We present a general form of the effective spin-chain model for strongly interacting atomic gases with an arbitrary spin in the one-dimensional(1D) traps. In particular, for high-spin systems the atoms can collide in multiple scattering channels, and we find that the resulted form of spin-chain model generically follows the same structure as that of the interaction potentials. This is a unified form working for any spin, statistics (Bose or Fermi) and confinement potentials. We adopt the spin-chain model to reveal both the ferromagnetic(FM) and anti-ferromagnetic(AFM) magnetic orders for strongly interacting spin-1 bosons in 1D traps. We further show that by adding the spin-orbit coupling, the FM/AFM orders can be gradually destroyed and eventually the ground state exhibits universal spin structure and contacts that are independent of the strength of spin-orbit coupling.