Competition between exchange and anisotropy in a pyrochlore ferromagnet

The Ising-like spin ice model, with a macroscopically degenerate ground state, has been shown to be approximated by several real materials. Here we investigate a model related to spin ice, in which the Ising spins are replaced by classical Heisenberg spins. These populate a cubic pyrochlore lattice and are coupled to nearest neighbours by a ferromagnetic exchange term J and to the local $\langle 111 \rangle$ axes by a single-ion anisotropy term D. The near-neighbour spin ice model corresponds to the case $D/J \rightarrow \infty$. For finite D/J we find that the macroscopic degeneracy of spin ice is broken and the ground state is magnetically ordered into a four-sublattice structure. The transition to this state is first order for D/J > 5 and second order for D/J < 5 with the two regions separated by a tricritical point. We investigate the magnetic phase diagram with an applied field along [001] and show that it can be considered analogous to that of a ferroelectric.