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 $\mathbf{k}=\{\frac{1}{2}\frac{1}{2}\frac{1}{2}\}$. Starting from a general microscopic Hamiltonian including anisotropic nearest-neighbor exchange, long-range dipolar interactions, and second- and third-nearest neighbor exchange, we use standard mean-field theory (SMFT) to identify an extended range of interaction parameters that support partially ordered phases. We demonstrate that thermal fluctuations ignored in SMFT are responsible for the selection of one particular partially ordered phase, e.g., the “$4\text{−}\mathbf{k}$” phase over the “$1\text{−}\mathbf{k}$” phase. We suggest that the transition into the $4\text{−}\mathbf{k}$ phase is continuous with its critical properties controlled by the cubic fixed point of a Ginzburg-Landau theory with a four-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 $4\text{−}\mathbf{k}$ partially ordered phase observed in the ${\mathrm{Gd}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ dipolar pyrochlore antiferromagnet below its paramagnetic phase transition temperature.

• Received 16 October 2013

DOI:https://doi.org/10.1103/PhysRevLett.114.130601