Structural phase transitions in geometrically frustrated antiferromagnets

We study geometrically frustrated antiferromagnets with magnetoelastic coupling. Frustration in these systems may be relieved by a structural transition to a low-temperature phase with reduced lattice symmetry. We examine the statistical mechanics of this transition and the effects on it of quenched disorder using Monte Carlo simulations of the classical Heisenberg model on the pyrochlore lattice with coupling to uniform lattice distortions. The model has a transition between a cubic paramagnetic high-temperature phase and a tetragonal Néel ordered low-temperature phase. It does not support the spin-Peierls phase, which is predicted as an additional possibility within Landau theory, and the transition is first order for reasons unconnected with the symmetry analysis of Landau theory. Quenched disorder stabilizes the cubic phase, and we find a phase diagram as a function of temperature and disorder strength similar to that observed in ${\text{Zn}}_{1-x}{\text{Cd}}_x{\text{Cr}}_2{\text{O}}_4$.