Complex disordered states—from liquids and glasses to exotic quantum matter—are ubiquitous in nature. Their key properties include finite entropy, power-law correlations and emergent organizing principles. In spin ice, spin correlations are determined by the ‘ice rules’ organizing principle that stabilizes a magnetic state with the same zero-point entropy as water ice. The entropy can be manipulated with great precision by an applied magnetic field: when directed along the three-fold crystallographic axis, the field produces a state of finite entropy, known as kagome ice. Here, we investigate the spin-ice material Ho2Ti2O7 by tilting the magnetic field slightly away from that axis. We thus realize a classic statistical system named after Kasteleyn, in which the entropy of a critical phase can be continuously tuned. Our neutron scattering experiments reveal how this process occurs at a microscopic level.
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Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. Geometric frustration in the ferromagnetic pyrochlore Ho2Ti2O7 . Phys. Rev. Lett. 79, 2554–2557 (1997).
Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. & Shastry, B. S. Zero-point entropy in ‘spin ice’. Nature 399, 333–335 (1999).
Snyder, J., Slusky, J. S., Cava, R. J. & Schiffer, P. How ‘spin ice’ freezes. Nature 413, 48–51 (2001).
Matsuhira, K., Hiroi, Z., Tayama, T., Takagi, S. & Sakakibara, T. A new macroscopically degenerate ground state in the spin ice compound Dy2Ti2O7 . J. Phys. Condens. Matter 14, L559–L565 (2002).
Sakakibara, T., Tayama, T., Hiroi, Z., Matsuhira, K. & Takagi, S. Observation of a liquid-gas-type transition in the pyrochlore spin ice compound Dy2Ti2O7 in a magnetic field. Phys. Rev. Lett. 90, 207205 (2003).
Higashinaka, R. & Maeno, Y. Field-induced transition of a triangular plane in the spin-ice compound Dy2Ti2O7 . Phys. Rev. Lett. 95, 237208 (2005).
Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc. 57, 2680–2684 (1935).
Isakov, S. V., Raman, K. S., Moessner, R. & Sondhi, S. L. Magnetization curve of spin ice in a  magnetic field. Phys. Rev. B 70, 104418 (2004).
Aoki, H., Sakakibara, T., Matsuhira, K. & Hiroi, Z. Magnetocaloric effect study on the pyrochlore spin ice compound Dy2Ti2O7 in a  magnetic field. J. Phys. Soc. Japan 73, 2851–2856 (2004).
Bonner, J. C. & Fisher, M. E. Entropy of an antiferromagnet in a magnetic field. Proc. Phys. Soc. 80, 508–515 (1962).
Tabata, Y. et al. Kagome ice state in the dipolar spin ice Dy2Ti2O7 . Phys. Rev. Lett. 97, 257205 (2006).
den Hertog, B. C. & Gingras, M. J. P. Dipolar interactions and origin of spin ice in ising pyrochlore magnets. Phys. Rev. Lett. 84, 3430–3433 (2000).
Melko, R. G. & Gingras, M. J. P. Monte Carlo studies of the dipolar spin ice model. J. Phys. Condens. Matter 16, R1277–R1319 (2004).
Isakov, S. V., Gregor, K., Moessner, R. & Sondhi, S. L. Dipolar spin correlations in classical pyrochlore magnets. Phys. Rev. Lett. 93, 167204 (2004).
Isakov, S. V., Moessner, R. & Sondhi, S. L. Why spin ice obeys the ice rules. Phys. Rev. Lett. 95, 217201 (2005).
Anderson, P. W. Ordering and antiferromagnetism in ferrites. Phys. Rev. 102, 1008–1013 (1956).
Henley, C. L. Power-law spin correlations in pyrochlore antiferromagnets. Phys. Rev. B 71, 014424 (2005).
Youngblood, R. W. & Axe, J. D. Polarization fluctuations in ferroelectric models. Phys. Rev. B 23, 232–238 (1981).
Skalyo, J., Frazer, B. C. & Shirane, G. Ferroelectric mode motion in KD2PO4 . Phys. Rev. B 1, 278–286 (1970).
Harris, M. J., Zinkin, M. P., Tun, Z., Wanklyn, B. M. & Swainson, I. P. Magnetic structure of the spin-liquid state in a frustrated pyrochlore. Phys. Rev. Lett. 73, 189–192 (1994).
Ballou, R., Leliévre-Berna, E. & Fåk, B. Spin fluctuations in (Y0.97Sc0.03)Mn2: A geometrically frustrated, nearly antiferromagnetic, itinerant electron system. Phys. Rev. Lett. 76, 2125–2128 (1996).
Bramwell, S. T. et al. Spin correlations in Ho2Ti2O7 : A dipolar spin ice. Phys. Rev. Lett. 87, 047205 (2001).
Fennell, T. et al. Neutron scattering investigation of the spin ice state in Dy2Ti2O7 . Phys. Rev. B 70, 134408 (2004).
Moessner, R. & Sondhi, S. L. Theory of the  magnetization plateau in spin ice. Phys. Rev. B 68, 064411 (2003).
Kasteleyn, P. W. Dimer statistics and phase transitions. J. Math. Phys. 4, 287–293 (1963).
Yokoi, C. S. O., Nagle, J. F. & Salinas, S. R. Dimer pair correlations on the brick lattice. J. Stat. Phys. 44, 729–747 (1986).
Collins, M. F. Magnetic Critical Scattering (Oxford Univ. Press, Oxford, 1989).
Petrenko, O. A., Lees, M. R. & Balakrishnan, G. Magnetization process in the spin-ice compound Ho2Ti2O7 . Phys. Rev. B 68, 012406 (2003).
Fennell, T. et al. Neutron scattering studies of the spin ices Ho2Ti2O7 and Dy2Ti2O7 in applied magnetic field. Phys. Rev. B 72, 224411 (2005).
Hiroi, Z., Matsuhira, K. & Ogata, M. Ferromagnetic Ising spin chains emerging from the spin ice under magnetic field. J. Phys. Soc. Japan 72, 3045–3048 (2003).
Mirebeau, I. et al. Ordered spin ice state and magnetic fluctuations in Tb2Sn2O7 . Phys. Rev. Lett. 94, 246402 (2005).
Gardner, J. S. et al. Cooperative paramagnetism in the geometrically frustrated antiferromagnet Tb2Ti2O7 . Phys. Rev. Lett. 82, 1012–1015 (1999).
Lee, S.-H. et al. Emergent excitations in a geometrically frustrated antiferromagnet. Nature 418, 856–858 (2002).
Lau, G. C. et al. Zero-point entropy in stuffed spin-ice. Nature Phys. 2, 249–253 (2006).
Taguchi, Y., Oohara, Y., Yoshizawa, H., Nagaosa, N. & Tokura, Y. Spin chirality, Berry phase and anomalous Hall effect in a frustrated ferromagnet. Science 291, 2573–2576 (2001).
Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006).
Castro Neto, A. H., Pujol, P. & Fradkin, E. Ice: A strongly correlated proton system. Phys. Rev. B 74, 024302 (2006).
Limelette, P. et al. Universality and critical behaviour at the Mott transition. Science 302, 89–92 (2003).
Kutnjak, Z., Petzelt, J. & Blinc, R. The giant electromechanical response in ferroelectric relaxors as a critical phenomenon. Nature 441, 956–959 (2006).
Isakov, S. V., Wessel, S., Melko, R. G., Sengupta, K. & Kim, Y. B. Valence bond solids and their quantum melting in hard core bosons on the kagome lattice. Phys. Rev. Lett. 97, 147202 (2006).
Wanklyn, B. M. Flux growth of some complex oxide materials. J. Mater. Sci. 7, 813–821 (1972).
Bramwell, S. T., Field, M. N., Harris, M. J. & Parkin, I. P. Bulk magnetization of the heavy rare earth pyrochlores—a series of model frustrated magnets. J. Phys. Condens. Matter 12, 483–495 (2000).
It is a pleasure to acknowledge the sample environment team at ISIS (in particular R. Down and J. Keeping) and the ILL (J.-L. Ragazzoni and O. Losserand) and we would like to thank A. S. Wills, R. Moessner and P. C. W. Holdsworth for valuable discussions. We thank the EPSRC (UK) for financial support; work in London was also supported by a Wolfson Royal Society Research Merit Award.
The authors declare no competing financial interests.
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Fennell, T., Bramwell, S., McMorrow, D. et al. Pinch points and Kasteleyn transitions in kagome ice. Nature Phys 3, 566–572 (2007) doi:10.1038/nphys632
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