We study the ground state phases for a mixture of two atomic spin-1 Bose-Einstein condensates (BECs) in the presence of a weak magnetic (B-) field. The ground state is found to contain a broken-axisymmetry (BA) phase due to competitions among intra- and inter-species spin exchange interactions and the linear Zeeman shifts. This is in contrast to the case of a single species spin- 1 condensate, where the axisymmetry breaking results from competitions among the linear and quadratic Zeeman shifts and the intra-species ferromagnetic interaction. All other remaining ground state phases for the mixture are found to preserve axisymmetry. We further elaborate on the ground state phase diagram and calculate their Bogoliubov excitation spectra. For the BA phase, there exist three Goldstone modes attempting to restore the broken U(1) and SO(2) symmetries.
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