Flucutation driven selection at crticality: the case of multi-k partial order on the pyrochlore lattice

We study the problem of partially ordered phases with periodically arranged disordered sites on the pyrochlore lattice. The periodicity of the phases is characterized by one or more wave vectors $k = \{\frac{1}{2}\frac{1}{2}\frac{1}{2}\}$. Starting from a general microscopic Hamiltonian including anisotropic nearest-neighbor exchange, long-range dipolar interactions and second- and third-nearest neighbor exchange, we identify using standard mean-field theory (s-MFT) an extended range of interaction parameters that support partially ordered phases. We demonstrate that thermal fluctuations beyond 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 result reveals the origin of 4-$k$ phase observed bellow 1K in Gd2Ti2O7.