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We compare the behaviour of ferromagnetic and antiferromagnetic Ising-type spin models on the cubic pyrochlore lattice. With simple `up - down' Ising spins, the antiferromagnet is highly frustrated and the ferromagnet is not. However, such spin symmetry cannot be realized on the pyrochlore lattice, since it requires a unique symmetry axis, which is incompatible with the cubic symmetry. The only two-state spin symmetry which is compatible is that with four local anisotropy axes, which direct the spins to point in or out of the tetrahedral plaquettes of the pyrochlore lattice. We show how the local `in - out' magnetic anisotropy reverses the roles of the ferro- and antiferromagnetic exchange couplings with regard to frustration, such that the ferromagnet is highly frustrated and the antiferromagnet is not. The in - out ferromagnet is a magnetic analogue of the ice model, which we have termed the `spin ice model'. It is realized in the material . The up - down antiferromagnet is also an analogue of the ice model, albeit a less direct one, as originally shown by Anderson. Combining these results shows that the up - down spin models map onto the in - out spin models with the opposite sign of the exchange coupling. We present Monte Carlo simulations of the susceptibility for each model, and discuss their relevance to experimental systems.

The pyrochlore material Ho2Ti2O7 has been suggested to show "spin ice" behaviour. We present neutron scattering and specific heat results that establish unambiguously that Ho2Ti2O7 exhibits spin ice correlations at low temperature. Diffuse magnetic neutron scattering from Ho2Ti2O7 is found to be quite well described by a nearest neighbour spin ice model and very accurately described by a dipolar spin ice model. The heat capacity is well accounted for by the sum of a dipolar spin ice contribution and an expected nuclear spin contribution, known to exist in other Ho 3+ salts. These results settle the question of the nature of the low temperature spin correlations in Ho2Ti2O7 for which contradictory claims have been made.

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Er2Ti2O7 has been suggested to be a realization of the frustrated 111 XY pyrochlore lattice antiferromagnet, for which theory predicts fluctuation-induced symmetry breaking in a highly degenerate ground state manifold. We present a theoretical analysis of the classical model compared to neutron scattering experiments on the real material, both below and above TN =1.173(2) K. The model correctly predicts the ordered magnetic structure, suggesting that the real system has order stabilized by zero-point quantum fluctuations that can be modelled by classical spin wave theory. However, the model fails to describe the excitations of the system, which show unusual features.PACS numbers: 28.20. Cz, 75.25.+z An important aspect of condensed matter is the separation of energy scales, such that the minimization of one set of interactions may result in the frustration of another. A paradigm is the frustrated antiferromagnet, in which the local magnetic couplings between ions are frustrated by the crystal symmetry that the ions adopt. However, a systematic study of the rare earth pyrochlore titanates R 2 Ti 2 O 7 has shown that local antiferromagnetic bond frustration is neither a necessary, nor a sufficient condition for magnetic frustration 1,2,3,4,5 . Rather, it arises from the interplay, in the context of the crystal symmetry, of the principal terms in the spin Hamiltonian. In the case of R 2 Ti 2 O 7 , the main terms are single-ion anisotropy, exchange and dipolar coupling. Depending on the balance of these factors, one observes spin ice behavior (R = Ho, Dy) 1,2,3 , spin liquid behavior (R = Tb) 4 , and dipole induced partial order (R = Gd)5 . Such behavior is best classified in terms of the dominant 111 single-ion anisotropy that arises from the trigonal crystal electric field (CEF) at the rare earth site. For example, whereas the Heisenberg antiferromagnet has a spin liquid ground state 6 , the 111 Ising (dipolar) ferromagnet has a spin ice ground state 1,3,7 . There is thus a clear motivation to study models based on other simple anisotropies and their realization in the titanate series. In this Letter we study one such model -the 111 XY model antiferromagnet 8 -and its realization Er 2 Ti 2 O 7 9,10,11,12 . We consider the Hamiltonian:where the classical spins, S i , populate a face centered cubic array of corner sharing tetrahedra: the pyrochlore lattice. The spins are confined to easy XY planes by a local d i = 111 anisotropy, D < 0, and are coupled antiferromagnetically by exchange J < 0. This model was first studied in Ref.8 , where a discrete, but macroscopically degenerate, set of ground states was identified. At finite temperature thermal fluctuations were found to select an ordered state by the mechanism that Villain called "order by disorder"13 and a first order phase transition was observed in numerical simulations. The propagation vector of the ordered state was found to be k = 0, 0, 0 (henceforth "k = 0"), but the basis vectors of the magnetic structure were not determined. We have recently discovere...