Order induced by dipolar interactions in a geometrically frustrated antiferromagnet

We study the classical Heisenberg model for spins on a pyrochlore lattice interacting via long-range dipole-dipole forces and nearest-neighbor exchange. Antiferromagnetic exchange alone is known not to induce ordering in this system. We analyze low-temperature order resulting from the combined interactions, both by using a mean-field approach and by examining the energy cost of fluctuations about an ordered state. We discuss behavior as a function of the ratio of the dipolar and exchange interaction strengths and find two types of ordered phase. Below a certain value of this ratio, we find that the system orders in a four-sublattice N\'eel state. For interaction strengths above this critical ratio, the system orders with an incommensurate wave vector. We relate our results to the recent experimental work and reproduce and extend the theoretical calculations on the pyrochlore compound, ${\mathrm{Gd}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7},$ by N. P. Raju, M. Dion, M. J. P. Gingras, T. E. Mason, and J. E. Greedan, Phys. Rev. B 59, 14 489 (1999).