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Searching for Non-Abelian Phases in the Bose-Einstein Condensate of Dysprosium

The recently realized Bose-Einstein condensate of Dysprosium will become a spin-8 spinor condensate at ultralow magnetic fields. In such a high spin condensate, many phases with different symmetries can exist. Among them the most interesting ones are those with non-abelian point group symmetry. In this letter we discuss the variety of symmetry phases in a spin-8 condensate resulting from numerical solutions of the Hamiltonian. We show that these symmetries can be determined uniquely from the measurements of density population on each spin component in an ultralow magnetic field, together with the measurements of the collective modes in the zero-field limit. This method can also be applied to Bose-Einstein condensate of other magnetic atoms, such as Cr and Er.