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We find nearly all the exact ground states of a mixture of two species ofspin-1 atoms with both interspecies and intraspecies spin exchanges in absenceof a magnetic field. The quantum phase diagram in the three-dimensionalparameter space and its two-dimensional cross sections are described. Theboundaries where the ground states are either continuous or discontinuous aredetermined, with the latter identified as where quantum phase transitions takeplace. The two species are always disentangled if the interspecies spincoupling is ferromagnetic or zero. Quantum phase transitions occur when theinterspecies spin coupling varies between antiferromagtic and zero orferromagnetic while the two intraspecies spin couplings both remainferromagnetic. On the other hand, by tuning the interspecies spin coupling fromzero to antiferromagnetic and then back to zero, one can circumvent the quantumphase transition due to sign change of the intraspecies spin coupling of asingle species, which is spin-decoupled with the other species withferromagnetic intraspecies spin coupling. Overall speaking, interplay amonginterspecies and two intraspecies spin exchanges significantly enriches quantumphases of spinor atomic gases.

Three-dimensional quantum phase diagram of the exact ground states of a mixture of two species of spin-1 Bose gases with interspecies spin exchange