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.

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Neutron scattering and ac-susceptibility techniques have been performed on the spin ice material Ho 2 Ti 2 O 7 to study the spin relaxation processes in the 'hot' paramagnetic phase (T > 1 K). Neutron spin echo (NSE) proves that above T 15 K the spin dynamics are governed by a thermally activated single-ion process. At lower temperatures (T < 15 K) this cannot account for the spin dynamics found in ac-susceptibility measurements. It is inferred that a second, slower process, with a different thermal signature dominates. We suggest that this is a quantum-mechanical tunnelling process between different spin states separated by a large energy barrier.

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The gas of magnetic monopoles in spin ice is governed by one key parameter: the monopole chemical potential. A significant variation of this parameter could access hitherto undiscovered magnetic phenomena arising from monopole correlations, as observed in the analogous electrical Coulomb gas, like monopole dimerization, critical phase separation, or charge ordering. However, all known spin ices have values of chemical potential imposed by their structure and chemistry that place them deeply within the weakly correlated regime, where none of these interesting phenomena occur. Here we use high-pressure synthesis to create a new monopole host, Dy2Ge2O7, with a radically altered chemical potential that stabilizes a large fraction of monopole dimers. The system is found to be ideally described by the classic Debye–Huckel–Bjerrum theory of charge correlations. We thus show how to tune the monopole chemical potential in spin ice and how to access the diverse collective properties of magnetic monopoles.

Condensed matter in the low-temperature limit reveals exotic physics associated with unusual orders and excitations, with examples ranging from helium superfluidity 1 to magnetic monopoles in spin ice 2,3 . The far-from-equilibrium physics of such low-temperature states may be even more exotic, yet to access it in the laboratory remains a challenge. Here we demonstrate a simple and robust techniquethe 'magnetothermal avalanche quench'-and its use in the controlled creation of non-equilibrium populations of magnetic monopoles in spin ice at millikelvin temperatures. These populations are found to exhibit spontaneous dynamical effects that typify far-from-equilibrium systems and yet are captured by simple models. Our method thus opens new directions in the study of far-from-equilibrium states in spin ice and other exotic magnets.The normal way of controlling the temperature of a system is to connect it thermally to a second body with a much larger thermal mass, which then acts as a thermal reservoir. If it is desired to force thermal excitations out of equilibrium by a rapid temperature quench, then the simplest strategy would be to heat the sample, and then abruptly remove the heating so that the sample is cooled rapidly by the reservoir. However, direct heating of the sample will also tend to heat the reservoir, and this becomes a particular problem in low-temperature devices: for example, in a 3 He-4 He dilution refrigerator it may entail heating of the mixing chamber, and ultimately limit the speed of any thermal quench that can be practically achieved. Our technique gets round this problem by using the natural tendency of magnets to undergo magnetothermal 'avalanches' at low temperature [4][5][6][7] . It is illustrated in Fig. 1 and discussed further in Supplementary Section 1.4. The essential principle is that magnetic work done on the sample is abruptly converted into internal heat, which causes a sudden increase in temperature inside the sample (T int ). The sample then finds itself at a relatively high temperature but connected to a cold thermal bath. The ensuing quench is as efficient and rapid as possible as it involves minimal heating of the sample environment, which remains at the reservoir temperature, T .Magnetothermal avalanches typically occur at low temperature (T < 1 K), a regime also notable for the occurrence of exotic magnetic states based on long-range interactions, quantum effects and magnetic frustration [8][9][10][11][12][13][14] . Hence, the avalanche quench technique could be generally used to drive such systems out of equilibrium. We focus on the case of spin ice, a nearly ideal realization of a magnetic ice-type or vertex model 8 . The far-from-equilibrium physics of vertex models is a subject of great intrinsic interest 15 In spin-ice materials such as Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7 , the frustrated pyrochlore lattice geometry and local crystal field combines with a self-screening dipole-dipole interaction to give a local 'ice rule' that controls low-energy spin configurations 8,16...

Spin ice illustrates many unusual magnetic properties, including zero point entropy, emergent monopoles and a quasi liquid–gas transition. To reveal the quantum spin dynamics that underpin these phenomena is an experimental challenge. Here we show how crucial information is contained in the frequency dependence of the magnetic susceptibility and in its high frequency or adiabatic limit. The typical response of Dy2Ti2O7 spin ice indicates that monopole diffusion is Brownian but is underpinned by spin tunnelling and is influenced by collective monopole interactions. The adiabatic response reveals evidence of driven monopole plasma oscillations in weak applied field, and unconventional critical behaviour in strong applied field. Our results clarify the origin of the relatively high frequency response in spin ice. They disclose unexpected physics and establish adiabatic susceptibility as a revealing characteristic of exotic spin systems.

The series of magnetic rare-earth pyrochlore stannates R2Sn2O7 (R = rare earth, except Ce and Pm) have been investigated by powder susceptibility measurements down to T = 1.8 K. The results are compared with results for the analogous titanate series, which are well known frustrated magnets. Unlike the titanates, the whole series can be formed in the cubic pyrochlore structure. Possible experimental advantages of studying the stannates are discussed. PACS Nos.: 75.10Nr, 75.50Ee, 75.50Lk