  We study the problem of partially ordered phases with periodically arranged
disordered (paramagnetic) sites on the pyrochlore lattice, a network of
corner-sharing tetrahedra. The periodicity of these phases is characterized by
one or more wave vectors k=(1/2 1/2 1/2). Starting from a general microscopic
Hamiltonian including anisotropic nearest-neighbor exchange, long-range dipolar
interactions and second- and third-nearest neighbor exchange, we identify using
standard mean-field theory (s-MFT) an extended range of interaction parameters
that support partially ordered phases. We demonstrate that thermal fluctuations
ignored in s-MFT are responsible for the selection of one particular partially
ordered phase, e.g. the "4-k" phase over the "1-k" phase. We suggest that the
transition into the 4-k phase is continuous with its critical properties
controlled by the cubic fixed point of a Ginzburg-Landau theory with a
4-component vector order-parameter. By combining an extension of the
Thouless-Anderson-Palmer method originally used to study fluctuations in spin
glasses with parallel-tempering Monte-Carlo simulations, we establish the phase
diagram for different types of partially ordered phases. Our results elucidate
the long-standing puzzle concerning the origin of the 4-k partially ordered
phase observed in the Gd2Ti2O7 dipolar pyrochlore antiferromagnet below its
paramagnetic phase transition temperature.
