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We study the problem of partially ordered phases with periodically arrangeddisordered (paramagnetic) sites on the pyrochlore lattice, a network ofcorner-sharing tetrahedra. The periodicity of these phases is characterized byone or more wave vectors k=(1/2 1/2 1/2). Starting from a general microscopicHamiltonian including anisotropic nearest-neighbor exchange, long-range dipolarinteractions and second- and third-nearest neighbor exchange, we identify usingstandard mean-field theory (s-MFT) an extended range of interaction parametersthat support partially ordered phases. We demonstrate that thermal fluctuationsignored in s-MFT are responsible for the selection of one particular partiallyordered phase, e.g. the "4-k" phase over the "1-k" phase. We suggest that thetransition into the 4-k phase is continuous with its critical propertiescontrolled by the cubic fixed point of a Ginzburg-Landau theory with a4-component vector order-parameter. By combining an extension of theThouless-Anderson-Palmer method originally used to study fluctuations in spinglasses with parallel-tempering Monte-Carlo simulations, we establish the phasediagram for different types of partially ordered phases. Our results elucidatethe long-standing puzzle concerning the origin of the 4-k partially orderedphase observed in the Gd2Ti2O7 dipolar pyrochlore antiferromagnet below itsparamagnetic phase transition temperature.

Fluctuation-Driven Selection at Criticality in a Frustrated Magnetic System: the Case of Multiple-k Partial Order on the Pyrochlore Lattice