Field driven phases in the geometrically frustrated dipolar Heisenberg pyrochlore antiferromagnet Gd$_2$Ti$_2$O$_7$

The rare-earth pyrochlore gadolinium titanate, Gd$_2$Ti$_2$O$_7$, represents an excellent experimental realization of a Heisenberg antiferromagnet (AFM) in a frustrated geometry with weak long-range dipole-dipole interactions (approximately 20$\%$ of nearest neighbor AFM exchange). Experiments on Gd$_2$Ti$_2$O$_7$ in a magnetic field reveal a complex phase diagram associated with the breaking of spatial symmetries of the pyrochlore lattice as the field is applied along select symmetry directions. We study a model of classical Heisenberg spins (O(3) symmetry) on a pyrochlore lattice with exchange and dipolar interactions within mean-field theory. Using parameters relevant to the material system, we develop phase diagrams in finite magnetic fields. Our results our compared to experiments on Gd$_2$Ti$_2$O$_7$ (and Gd$_2$Sn$_2$O$_7$).