We have measured the specific heat of the coupled spin-dimer antiferromagnet Ba$_3$Mn$_2$O$_8$ to 50 mK in temperature and to 29 T in the magnetic field. The experiment extends to the midpoint of the field region (25.9 T $leq H leq$ 32.3 T) of the magnetization plateau at 1/2 of the saturation magnetization, and reveals the presence of three ordered phases in the field region between that of the magnetization plateau and the low-field spin-liquid region. The exponent of the phase boundary with the thermally disordered region is smaller than the theoretical value based on the Bose-Einstein condensation of spin triplets. At zero field and 29 T, the specific-heat data show gapped behaviors characteristic of spin liquids. The zero-field data indicate that the gapped triplet excitations form two levels whose energies differ by nearly a factor of two. At least the lower level is well localized. The data at 29 T reveal that the low-lying excitations at the magnetization plateau are weakly delocalized.
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