Ordering in a frustrated pyrochlore antiferromagnet proximate to a spin liquid

We perform a general study of spin ordering on the pyrochlore lattice with a 3:1 proportionality of two spin polarizations. Equivalently, this describes valence bond solid conformations of a quantum dimer model on the diamond lattice. We determine the set of likely low-temperature ordered phases, on the assumption that the ordering is weak, i.e., the system is close to a U(1) quantum spin liquid in which the 3:1 proportionality is maintained but the spins are strongly fluctuating. The nature of the nine ordered states we find is determined by a projective symmetry analysis. All the phases exhibit translational and rotational symmetry breaking, with an enlarged unit cell containing 4-64 primitive cells of the underlying pyrochlore. The simplest of the nine phases is the same R state found earlier in a theoretical study of the ordering on the magnetization plateau in the S=3/2 materials CdCr2O4 and HgCr2O4. We suggest that the spin-dimer model proposed therein undergoes a direct transition from the spin liquid to the R state, and describe a field theory for the universal properties of this critical point, at zero and nonzero temperatures.