- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 3.0 License. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice

L. D. C. Jaubert1,2,3,4,*, M. J. Harris5, T. Fennell6, R. G. Melko7,8, S. T. Bramwell9, and P. C. W. Holdsworth1

  • 1Laboratoire de Physique, École Normale Supérieure de Lyon, Université de Lyon, CNRS, 46 Allée d’Italie, 69364 Lyon Cedex 07, France
  • 2Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany
  • 3Theoretical Physics, Oxford University, Oxford, OX1 3NP, United Kingdom
  • 4OIST—Okinawa Institute of Science and Technology, Onna-son, Okinawa 904-0495, Japan
  • 5School of Divinity, University of Edinburgh, New College, Mound Place, Edinburgh, EH1 2LX, United Kingdom
  • 6Paul Scherrer Institut, 5232 Villigen PSI, Switzerland
  • 7Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
  • 8Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
  • 9London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, 17-19 Gordon Street, London WC1H 0AH, United Kingdom

  • *ludovic.jaubert@oist.jp
PDF

Phys. Rev. X 3, 011014 – Published 21 February, 2013

DOI: https://doi.org/10.1103/PhysRevX.3.011014

Abstract

At low temperatures, a spin ice enters a Coulomb phase—a state with algebraic correlations and topologically constrained spin configurations. We show how analytical and numerical approaches for model spin-ice systems reveal a crossover between two Curie laws. One of these laws characterizes the high-temperature paramagnetic regime, while the other, which we call the “spin-liquid Curie law,” characterizes the low-temperature Coulomb-phase regime, which provides implicit evidence that the topological sector fluctuates. We compare our theory with experiment for Ho2Ti2O7, where this process leads to a nonstandard temperature evolution of the bulk susceptibility and the wave-vector-dependent magnetic susceptibility, as measured by neutron scattering. Theory and experiment agree for bulk quantities and at large scattering wave vectors, but differences at small wave vectors indicate that the classical spin-ice states are not equally populated at low temperatures. More generally, the crossover appears to be a generic property of the emergent gauge field for a classical spin liquid, and it sheds light on the experimental difficulty of measuring a precise Curie-Weiss temperature in frustrated materials. The susceptibility at finite wave vectors is shown to be a local probe of fluctuations among topological sectors on varying length scales.

View figure in article

Popular Summary

In many situations in life, frustration is counterproductive and best avoided. In physics, however, frustration—the presence of competing forces that cannot be simultaneously satisfied—is often the source of exotic physical properties of materials systems. The exotic spin-liquid behavior in frustrated magnets provides just such an example. Defying the conventional expectation that at low temperatures individual microscopic spins in a material should become ordered in their orientations, spins in a frustrated magnet are thought to show no apparent orientational order. Rather, their orientations fluctuate among a large number of configurations where the fluctuations, though order-destroying, are actually highly correlated. Adding more kindling to the fire of interest in this so-called spin-liquid state are the tantalizing possibilities that spin liquids can host a range of very exotic phenomena, not least, topological phases and artificial gauge fields. In this article, we report a combined theoretical and experimental study of a frustrated magnet (Ho2Ti2O7) that points to the presence of a low-temperature spin-liquid state in which the correlations in the spin fluctuations are topological in nature.

Ho2Ti2O7 is a well-known frustrated magnet. In this material, two competing forces frustrate each other: While the physical interaction between the individual microscopic spins requires the spins to align in the same direction, the geometry of the crystalline lattice the spins are located in makes it impossible for this requirement to be met for all spins. The result is a local spin-lattice structure that resembles the local coordinated oxygen framework in water, ice, and a large number of low-energy, but energetically equivalent states of globally correlated spin configurations. These configurations can be classified into different sectors by a topological invariant. We are able to show that experimental techniques such as neutron scattering and magnetic susceptibility measurements, which probe spin-spin correlations, can be used to gauge the topological properties. Indeed, we have succeeded in identifying an abnormal temperature dependence in the spin-spin correlations as the emergence of correlated fluctuations among the many topological sectors at very low temperatures, indicative of a spin-liquid state characterized by topological fluctuations.

We believe that our findings are general, in that they should apply to a vast family of models and materials for frustrated magnets.

Article Text

References (87)

  1. A. P. Ramirez, Quantum Spin Liquids: A Flood or a Trickle?, Nat. Phys. 4, 442 (2008).
  2. P. A. Lee, An End to the Drought of Quantum Spin Liquids, Science 321, 1306 (2008).
  3. L. Savary and L. Balents, Coulombic Quantum Liquids in Spin-1/2 Pyrochlores, Phys. Rev. Lett. 108, 037202 (2012).
  4. J. D. Thompson, P. A. McClarty, H. M. Ronnow, L. P. Regnault, A. Sorge, and M. J. P. Gingras, Rods of Neutron Scattering Intensity in Yb2Ti2O7: Compelling Evidence for Significant Anisotropic Exchange in a Magnetic Pyrochlore Oxide, Phys. Rev. Lett. 106, 187202 (2011).
  5. L.-J. Chang, S. Onoda, Y. Su, Y.-J. Kao, K.-D. Tsuei, Y. Yasui, K. Kakurai, and M. R. Lees, Higgs Transition from a Magnetic Coulomb Liquid to a Ferromagnet in Yb2Ti2O7, Nat. Commun. 3, 992 (2012).
  6. R. Applegate, N. Hayre, R. Singh, T. Lin, A. Day, and M. Gingras, Vindication of Yb2Ti2O7 as a Model Exchange Quantum Spin Ice, Phys. Rev. Lett. 109, 097205 (2012).
  7. E. Kermarrec, P. Mendels, F. Bert, R. H. Colman, A. S. Wills, P. Strobel, P. Bonville, A. Hillier, and A. Amato, Spin-Liquid Ground State in the Frustrated Kagome Antiferromagnet MgCu3(OH)6Cl2, Phys. Rev. B 84, 100401 (2011).
  8. B. Fåk, E. Kermarrec, L. Messio, B. Bernu, C. Lhuillier, F. Bert, P. Mendels, B. Koteswararao, F. Bouquet, J. Ollivier et al., Kapellasite: A Kagome Quantum Spin Liquid with Competing Interactions, Phys. Rev. Lett. 109, 037208 (2012).
  9. L. Messio, B. Bernu, and C. Lhuillier, Kagome Antiferromagnet: A Chiral Topological Spin Liquid?, Phys. Rev. Lett. 108, 207204 (2012).
  10. T. Han, S. Chu, and Y. S. Lee, Refining the Spin Hamiltonian in the Spin-12 Kagome Lattice Antiferromagnet ZnCu3(OH)6Cl2 Using Single Crystals, Phys. Rev. Lett. 108, 157202 (2012).
  11. M. Hermele, M. P. A. Fisher, and L. Balents, Pyrochlore Photons: The U(1) Spin Liquid in a S=1/2 Three-Dimensional Frustrated Magnet, Phys. Rev. B 69, 064404 (2004).
  12. K. Ross, L. Savary, B. D. Gaulin, and L. Balents, Quantum Excitations in Quantum Spin Ice, Phys. Rev. X 1, 21002 (2011).
  13. O. Benton, O. Sikora, and N. Shannon, Seeing the Light: Experimental Signatures of Emergent Electromagnetism in a Quantum Spin Ice, Phys. Rev. B 86, 075154 (2012).
  14. N. Shannon, O. Sikora, F. Pollmann, K. Penc, and P. Fulde, Quantum Ice: A Quantum Monte Carlo Study, Phys. Rev. Lett. 108, 067204 (2012).
  15. L. Balents, Spin Liquids in Frustrated Magnets, Nature (London) 464, 199 (2010).
  16. C. Castelnovo, R. Moessner, S. L. Sondhi, Spin Ice, Fractionalization, and Topological Order, Annu. Rev. Condens. Matter Phys. 3, 35 (2012).
  17. X. G. Wen, Quantum Orders and Symmetric Spin Liquids, Phys. Rev. B 65, 165113 (2002).
  18. M. Levin and X. G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96, 110405 (2006).
  19. A. Kitaev and J. Preskill, Topological Entanglement Entropy, Phys. Rev. Lett. 96, 110404 (2006).
  20. Y. Ran, P. Hosur, and A. Vishwanath, Fermionic Hopf Solitons and Berry Phase in Topological Surface Superconductors, Phys. Rev. B 84, 184501 (2011).
  21. M. Freedman, M. B. Hastings, C. Nayak, and X.-L. Qi, Weakly Coupled Non-Abelian Anyons in Three Dimensions, Phys. Rev. B 84, 245119 (2011).
  22. A. J. Macdonald, P. C. W. Holdsworth, and R. G. Melko, Classical Topological Order in Kagome Ice, J. Phys. Condens. Matter 23, 164208 (2011).
  23. B. Canals and D. A. Garanin, Spin-Liquid Phase in the Pyrochlore Anti-ferromagnet, Can. J. Phys. 79, 1323 (2001).
  24. S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, Dipolar Spin Correlations in Classical Pyrochlore Magnets, Phys. Rev. Lett. 93, 167204 (2004).
  25. C. L. Henley, Power-Law Spin Correlations in Pyrochlore Antiferromagnets, Phys. Rev. B 71, 014424 (2005).
  26. P. H. Conlon and J. T. Chalker, Absent Pinch Points and Emergent Clusters: Further Neighbor Interactions in the Pyrochlore Heisenberg Antiferromagnet, Phys. Rev. B 81, 237206 (2010).
  27. M. P. Zinkin, M. J. Harris, and T. Zeiske, Short-Range Magnetic Order in the Frustrated Pyrochlore Antiferromagnet CsNiCrF6, Phys. Rev. B 56, 11786 (1997).
  28. T. Fennell, S. T. Bramwell, D. F. McMorrow, P. Manuel, and A. R. Wildes, Pinch Points and Kasteleyn Transitions in Kagome Ice, Nat. Phys. 3, 566 (2007).
  29. T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prabhakaran, A. T. Boothroyd, R. J. Aldus, D. F. McMorrow, and S. T. Bramwell, Magnetic Coulomb Phase in the Spin Ice Ho2Ti2O7, Science 326, 415 (2009).
  30. C. L. Henley, The “Coulomb Phase” in Frustrated Systems, Annu. Rev. Condens. Matter Phys. 1, 179 (2010).
  31. S. T. Bramwell and M. J. Harris, Frustration in Ising-Type Spin Models on the Pyrochlore Lattice, J. Phys. Condens. Matter 10, L215 (1998).
  32. H. Kadowaki, N. Doi, Y. Aoki, Y. Tabata, T. J. Sato, J. W. Lynn, K. Matsuhira, and Z. Hiroi, Observation of Magnetic Monopoles in Spin Ice, J. Phys. Soc. Jpn. 78, 103706 (2009).
  33. L. J. Chang, Y. Su, Y.-J. Kao, Y. Z. Chou, R. Mittal, H. Schneider, T. Brückel, G. Balakrishnan, and M. R. Lees, Magnetic Correlations in the Spin Ice Ho2−xYxTi2O7 as Revealed by Neutron Polarization Analysis, Phys. Rev. B 82, 172403 (2010).
  34. S. T. Bramwell, M. J. Harris, B. C. den Hertog, M. J. P. Gingras, J. S. Gardner, D. F. McMorrow, A. R. Wildes, A. L. Cornelius, J. D. M. Champion, R. G. Melko et al., Spin Correlations in Ho2Ti2O7: A Dipolar Spin Ice System, Phys. Rev. Lett. 87, 047205 (2001).
  35. M. J. Harris and S. T. Bramwell (unpublished).
  36. P. W. Anderson, Ordering and Antiferromagnetism in Ferrites, Phys. Rev. 102, 1008 (1956).
  37. R. Moessner, Relief and Generation of Frustration in Pyrochlore Magnets by Single-Ion Anisotropy, Phys. Rev. B 57, R5587 (1998).
  38. L. Pauling, The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement, J. Am. Chem. Soc. 57, 2680 (1935).
  39. A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan, and B. S. Shastry, Zero-Point Entropy in “Spin Ice”, Nature (London) 399, 333 (1999).
  40. H. Cao, A. Gukasov, I. Mirebeau, P. Bonville, C. Decorse, and G. Dhalenne, Ising Versus XY Anisotropy in Frustrated R2Ti2O7 Compounds as Seen by Polarized Neutrons, Phys. Rev. Lett. 103, 056402 (2009).
  41. L. D. C. Jaubert, J. T. Chalker, P. C. W. Holdsworth, and R. Moessner, Three-Dimensional Kasteleyn Transition: Spin Ice in a [100] Field, Phys. Rev. Lett. 100, 067207 (2008).
  42. S. Powell and J. T. Chalker, Classical to Quantum Mappings for Geometrically Frustrated Systems: Spin-Ice in a [100] Field, Phys. Rev. B 78, 024422 (2008).
  43. S. Powell, Higgs Transitions of Spin Ice, Phys. Rev. B 84, 094437 (2011).
  44. S. V. Isakov, K. S. Raman, R. Moessner, and S. L. Sondhi, Magnetization Curve of Spin Ice in a [111] Magnetic Field, Phys. Rev. B 70, 104418 (2004).
  45. Also called a loop algorithm by Melko et al. [46].
  46. R. G. Melko and M. J. P. Gingras, Monte Carlo Studies of the Dipolar Spin Ice Model, J. Phys. Condens. Matter 16, R1277 (2004).
  47. R. Moessner and S. L. Sondhi, Theory of the [111] Magnetization Plateau in Spin Ice, Phys. Rev. B 68, 064411 (2003).
  48. G. I. Watson, Symmetry Relations for the Six-Vertex Model, J. Stat. Phys. 94, 1045 (1999).
  49. D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J. U. Hoffmann et al., Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7, Science 326, 411 (2009).
  50. C. Castelnovo, R. Moessner, and S. L. Sondhi, Magnetic Monopoles in Spin Ice, Nature (London) 451, 42 (2008).
  51. I. A. Ryzhkin, Magnetic Relaxation in Rare-Earth Oxide Pyrochlores, J. Exp. Theor. Phys. 101, 481 (2005).
  52. L. D. C. Jaubert and P. C. W. Holdsworth, Signature of Magnetic Monopole and Dirac String Dynamics in Spin Ice, Nat. Phys. 5, 258 (2009).
  53. C. Castelnovo and C. Chamon, Topological Order and Topological Entropy in Classical Systems, Phys. Rev. B 76, 174416 (2007).
  54. S. Yoshida, K. Nemoto, and K. Wada, Application of the Cluster Variation Method to Spin Ice Systems on the Pyrochlore Lattice, J. Phys. Soc. Jpn. 71, 948 (2002).
  55. L. D. C. Jaubert, Topological Constraints and Defects in Spin Ice, Ph.D. thesis, Ecole Normale Supérieure de Lyon, 2009.
  56. L. D. C. Jaubert and P. C. W. Holdsworth, Curie Law Crossover in Frustrated Systems (unpublished).
  57. B. C. den Hertog and M. J. P. Gingras, Dipolar Interactions and Origin of Spin Ice in Ising Pyrochlore Magnets, Phys. Rev. Lett. 84, 3430 (2000).
  58. S. V. Isakov, R. Moessner, and S. L. Sondhi, Why Spin Ice Obeys the Ice Rules, Phys. Rev. Lett. 95, 217201 (2005).
  59. S. T. Bramwell, M. N. Field, M. J. Harris, and I. P. Parkin, Bulk Magnetization of the Heavy Rare Earth Titanate Pyrochlores—A Series of Model Frustrated Magnets, J. Phys. Condens. Matter 12, 483 (2000).
  60. A. P. Ramirez, Strongly Geometrically Frustrated Magnets, Annu. Rev. Mater. Sci. 24, 453 (1994).
  61. J. P. Clancy, J. P. C. Ruff, S. R. Dunsiger, Y. Zhao, H. A. Dabkowska, J. S. Gardner, Y. Qiu, J. R. D. Copley, T. Jenkins, and B. D. Gaulin, Revisiting Static and Dynamic Spin-Ice Correlations in Ho2Ti2O7 with Neutron Scattering, Phys. Rev. B 79, 014408 (2009).
  62. See, for example, Eq. 4.12 in H. Kadowaki, Y. Ishii, K. Matsuhira, and Y. Hinatsu, Neutron Scattering Study of Dipolar Spin Ice Ho2Sn2O7: Frustrated Pyrochlore Magnet, Phys. Rev. B 65, 144421 (2002).
  63. R. W. Youngblood and J. D. Axe, Polarization Fluctuations in Ferroelectric Models, Phys. Rev. B 23, 232 (1981).
  64. S. M. Bhattacharjee, J. F. Nagle, D. A. Huse, and M. E. Fisher, Critical-Behavior of a Three-Dimensional Dimer Model, J. Stat. Phys. 32, 361 (1983).
  65. L. D. C. Jaubert, M. Haque, and R. Moessner, Analysis of a Fully Packed Loop Model Arising in a Magnetic Coulomb Phase, Phys. Rev. Lett. 107, 177202 (2011).
  66. A. Sen, R. Moessner, and S. L. Sondhi, arXiv:1212.2112.
  67. T. Yavors’kii, T. Fennell, M. J. P. Gingras, and S. T. Bramwell, Dy2Ti2O7 Spin Ice: A Test Case for Emergent Clusters in a Frustrated Magnet, Phys. Rev. Lett. 101, 037204 (2008).
  68. R. G. Melko, B. C. den Hertog, and M. J. P. Gingras, Long-Range Order at Low Temperatures in Dipolar Spin Ice, Phys. Rev. Lett. 87, 067203 (2001).
  69. O. Sikora, F. Pollmann, N. Shannon, K. Penc, and P. Fulde, Quantum Liquid with Deconfined Fractional Excitations in Three Dimensions, Phys. Rev. Lett. 103, 247001 (2009).
  70. O. Sikora, N. Shannon, F. Pollmann, K. Penc, and P. Fulde, Extended Quantum U(1)-Liquid Phase in a Three-Dimensional Quantum Dimer Model, Phys. Rev. B 84, 115129 (2011).
  71. T. Fennell et al. (unpublished).
  72. J. Snyder, B. G. Ueland, J. S. Slusky, H. Karunadasa, R. J. Cava, and P. Schiffer, Low-Temperature Spin Freezing in the Dy2Ti2O7 Spin Ice, Phys. Rev. B 69, 064414 (2004).
  73. K. Matsuhira, Y. Hinatsu, K. Tenya, and T. Sakakibara, Low Temperature Magnetic Properties of Frustrated Pyrochlore Ferromagnets Ho2Sn2O7 and Ho2Ti2O7, J. Phys. Condens. Matter 12, L649 (2000).
  74. E. Lhotel, C. Paulsen, P. Dalmas de Réotier, A. Yaouanc, C. Marin, and S. Vanishri, Low-Temperature Magnetization in Geometrically Frustrated Tb2Ti2O7, Phys. Rev. B 86, 020410 (2012).
  75. G. Ehlers, A. L. Cornelius, M. Orendac, M. Kajnakova, T. Fennell, S. T. Bramwell, and J. S. Gardner, Dynamical Crossover in ’Hot’ Spin Ice, J. Phys. Condens. Matter 15, L9 (2003).
  76. L. R. Yaraskavitch, H. M. Revell, S. Meng, K. A. Ross, H. M. L. Noad, H. A. Dabkowska, B. D. Gaulin, and J. B. Kycia, Spin Dynamics in the Frozen State of the Dipolar Spin Ice Material Dy2Ti2O7, Phys. Rev. B 85, 020410 (2012).
  77. D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi, Coulomb and Liquid Dimer Models in Three Dimensions, Phys. Rev. Lett. 91, 167004 (2003).
  78. G. Misguich, B. Bernu, and L. Pierre, Determination of the Exchange Energies in Li2VOSiO4 from a High-Temperature Series Analysis of the Square-Lattice J(1)−J(2) Heisenberg Model, Phys. Rev. B 68, 113409 (2003).
  79. R. Moessner, Magnets with Strong Geometric Frustration, Can. J. Phys. 79, 1283 (2001).
  80. P. Mendels, F. Bert, M. A. de Vries, A. Olariu, A. Harrison, F. Duc, J. C. Trombe, J. S. Lord, A. Amato, and C. Baines, Quantum Magnetism in the Paratacamite Family: Towards an Ideal Kagome Lattice, Phys. Rev. Lett. 98, 077204 (2007).
  81. J. S. Gardner, A. Keren, G. Ehlers, C. Stock, E. Segal, J. M. Roper, B. Fak, M. B. Stone, P. R. Hammar, D. H. Reich et al., Dynamic Frustrated Magnetism in Tb2Ti2O7 at 50 mK, Phys. Rev. B 68, 180401 (2003).
  82. J. S. Gardner, M. J. P. Gingras, and J. E. Greedan, Magnetic Pyrochlore Oxides, Rev. Mod. Phys. 82, 53 (2010).
  83. T. Fennell, M. Kenzelmann, B. Roessli, M. K. Haas, and R. J. Cava, Power-Law Spin Correlations in the Pyrochlore Antiferromagnet Tb2Ti2O7, Phys. Rev. Lett. 109, 017201 (2012).
  84. J. D. Thompson, P. A. McClarty, and M. J. P. Gingras, Local Susceptibility of the Yb2Ti2O7 Rare Earth Pyrochlore Computed from a Hamiltonian with Anisotropic Exchange, J. Phys. Condens. Matter 23, 164219 (2011).
  85. H. Ju, A. B. Kallin, P. Fendley, M. B. Hastings, and R. G. Melko, Entanglement Scaling in Two-Dimensional Gapless Systems, Phys. Rev. B 85, 165121 (2012).
  86. S. T. Bramwell and M. J. P. Gingras, Spin Ice State in Frustrated Magnetic Pyrochlore Materials, Science 294, 1495 (2001).
  87. S. Rosenkranz, A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan, and B. S. Shastry, Crystal-Field Interaction in the Pyrochlore Magnet Ho2Ti2O7, J. Appl. Phys. 87, 5914 (2000).

Outline

Information

Phys. Rev. X 3, 011014– Published 21 February, 2013
  • Received 27 April 2012
Creative Commons Logo

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation