Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 499–514 | Cite as

Artificial Spin-Ice and Vertex Models

Article

Abstract

In classical and quantum frustrated magnets the interactions in combination with the lattice structure impede the spins to order in optimal configurations at zero temperature. The theoretical interest in their classical realisations has been boosted by the artificial manufacture of materials with these properties, that are of flexible design. This note summarises work on the use of vertex models to study bidimensional spin-ices samples, done in collaboration with R. A. Borzi, M. V. Ferreyra, L. Foini, G. Gonnella, S. A. Grigera, P. Guruciaga, D. Levis, A. Pelizzola and M. Tarzia, in recent years. It is an invited contribution to a J. Stat. Mech. special issue dedicated to the memory of Leo P. Kadanoff.

Keywords

Frustrated magnetism Vertex models Artificial spin-ice 

Notes

Acknowledgements

The research described in this note was carried on between 2011 and 2016 in collaboration with R. A. Borzi, M. V. Ferreyra, L. Foini, G. Gonnella, S. A. Grigera, P. C. Guruciaga, D. Levis, A. Pelizzola, J. Restrepo and M. Tarzia, all of whom I warmly thank. I also want to thank P. Holdsworth and L. Jaubert for very useful discussions and suggestions. I acknowledge financial support from ECOS-Sud A14E01, PICS 506691 (CNRS-CONICET Argentina) and NSF under Grant No. PHY11-25915. I am a member of Institut Universitaire de France.

References

  1. 1.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Dover, London (1982)MATHGoogle Scholar
  2. 2.
    Lieb, E.H., Wu, F.Y.: Two dimensional ferroelectric models. In: Domb, C., Green, M. (eds.) Phase Transitions and Critical Phenomena, pp. 331–490. Academic Press, London (1972)Google Scholar
  3. 3.
    Reshetikhin, N.: Lectures on the integrability of the six vertex model. In: Jacobsen, J., Ouvry, S., Pasquier, V., Serban, D., Cugliandolo, L.F. (eds.) Exact Methods in Low Dimensional Statistical Physics and Quantum Computing. Les Houches Summer School, vol. 89. Oxford University Press, Oxford (2008)Google Scholar
  4. 4.
    Marrows, C.: Experimental studies of artificial spin-ice (2016). arXiv:1611.00744
  5. 5.
    Levis, D.: Two dimensional spin-ice and the sixteen-vertex model. PhD thesis, Université Pierre et Marie Curie, Paris, France (2012)Google Scholar
  6. 6.
    Levis, D., Cugliandolo, L.F., Foini, L., Tarzia, M.: Thermal phase transitions in artificial spin ice. Phys. Rev. Lett. 110, 207206 (2013)ADSCrossRefGoogle Scholar
  7. 7.
    Foini, L., Levis, D., Tarzia, M., Cugliandolo, L.F.: Static properties of 2D spin-ice as a sixteen-vertex model. J. Stat. Mech. 2013(2), P02026 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Levis, D., Cugliandolo, L.F.: Quench dynamic of 2d spin ice models. EPL 97, 30002 (2012)ADSCrossRefGoogle Scholar
  9. 9.
    Levis, D., Cugliandolo, L.F.: Defects dynamics following thermal quenches in square spin ice. Phys. Rev. B 87, 214302 (2013)ADSCrossRefGoogle Scholar
  10. 10.
    Cugliandolo, L.F., Gonnella, G., Pelizzola, A.: Six-vertex model with domain wall boundary conditions in the Bethe-Peierls approximation. J. Stat. Mech. 2015(6), P06008 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Guruciaga, P.C., et al.: Field-tuned order by disorder in frustrated Ising magnets with antiferromagnetic interactions. Phys. Rev. Lett. 117, 167203 (2016)ADSCrossRefGoogle Scholar
  12. 12.
    Lieb, E.H.: Exact solution of the problem of the entropy of two-dimensional ice. Phys. Rev. Lett. 18, 692 (1967)ADSCrossRefGoogle Scholar
  13. 13.
    Lieb, E.H.: Exact solution of the F model of an antiferroelectric. Phys. Rev. Lett. 18, 1046 (1967)ADSCrossRefGoogle Scholar
  14. 14.
    Lieb, E.H.: Exact solution of the two-dimensional Slater KDP model of a ferroelectric. Phys. Rev. Lett. 19, 108 (1967)ADSCrossRefGoogle Scholar
  15. 15.
    Sutherland, B.: Exact solution of a two-dimensional model for hydrogen-bonded crystals. Phys. Rev. Lett. 19, 103 (1967)ADSCrossRefGoogle Scholar
  16. 16.
    Bramwell, S.T., et al.: Frustrated Spin Systems. World Scientific, Singapore (2004)Google Scholar
  17. 17.
    Balents, L.: Spin liquids in frustrated magnets. Nature 464, 199–208 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Gingras, M.J.P.: Spin ice. In: Lacroix, F.M.C., Mendels, P. (eds.) Highly Frustrated Magnetism. Springer, Berlin (2010)Google Scholar
  19. 19.
    Harris, M.J., Bramwell, S.T., Mcmorrow, D.F., Zeiske, T., Godfrey, K.W.: Geometrical frustration in the ferromagnetic pyrochlore \(\text{ Ho }_2\text{ Ti }_2\text{ O }_7\). Phys. Rev. Lett. 79, 2554–2557 (1997)ADSCrossRefGoogle Scholar
  20. 20.
    Bernal, J.D., Fowler, R.H.: A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions. J. Chem. Phys. 1, 515 (1933)ADSCrossRefGoogle Scholar
  21. 21.
    Ramírez, A.P., Hayashi, A., Cava, R.J., Siddharthan, R.S., Shastry, B.S.: Zero-point entropy in ‘spin ice’. Nature 399, 333 (1999)ADSCrossRefGoogle Scholar
  22. 22.
    Bramwell, S.T., Gingras, M.J.: Spin ice state in frustrated magnetic pyrochlore materials. Science (New York, N.Y.) 294(1495), 1495–501 (2001)ADSCrossRefGoogle Scholar
  23. 23.
    Giauque, W.F., Stout, J.W.: The entropy of water and the Third Law of Thermodynamics. The heat capacity of ice from 15 to 273 K. J. Am. Chem. Soc. 58, 1144–1150 (1936)ADSCrossRefGoogle Scholar
  24. 24.
    Pauling, L.: The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Chem. Phys. 57, 2680–2684 (1935)Google Scholar
  25. 25.
    Izergin, A.: Partition function of the 6-vertex model in a finite volume. Sov. Phys. Dokl. 32, 878 (1987)ADSMathSciNetMATHGoogle Scholar
  26. 26.
    Kuperberg, G.: Another proof of the alternative-sign matrix conjecture. Int. Math. Res. Notes 3, 139–150 (1996)CrossRefMATHGoogle Scholar
  27. 27.
    Korepin, V., Zinn-Justin, P.: Thermodynamic limit of the six-vertex model with domain wall boundary conditions. J. Phys. A 33, 7053 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Elkies, N., Kuperberg, G., Larsen, M., Propp, J.: Alternating sign matrices and domino tilings. J. Algebr. Comb. 1, 111 (1992)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Elkies, N., Kuperberg, G., Larsen, M., Propp, J.: Alternating sign matrices and domino tilings. J. Algebr. Comb. 1, 219 (1992)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Cohn, H., Elkis, N., Propp, J.: Local statistics of random domino tilings of the Aztec diamonds. Duke Math J. 85, 117 (1996)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jockusch, W, Propp, J., Shor, P.: Random domino tilings and the arctic circle theorem. arXiv:math.CO/9801088
  32. 32.
    Cohn, H., Marsen, L., Propp, J.: The shape of a typical boxed plane partition. New York J. Math. 4, 137 (1998)MathSciNetMATHGoogle Scholar
  33. 33.
    Borodin, A., Gorin, V., Rains, E.M.: Rains, q-distributions on boxed plane partitions. Selecta Math. (N.S.)16(4), 731–789 (2010)Google Scholar
  34. 34.
    Kenyon, R.: Lectures on dimers. In: Jacobsen, J., Ouvry, S., Pasquier, V., Serban, D., Cugliandolo, L.F. (eds.) Exact Methods in Low Dimensional Statistical Physics and Quantum Computing. Les Houches Summer School, vol. 89. Oxford University Press, Oxford (2008)Google Scholar
  35. 35.
    Baik, J., Kriecherbauer, T., Li, L.C., McLaughlin, K.T.R., Tomei, C. eds: Conference on Integrable Systems, Random Matrices, and Applications. The Arctic curve revisited. Contemporary Mathematics, vol. 458 (2008)Google Scholar
  36. 36.
    Colomo, F., Pronko, A.G., Zinn-Justin, P.: The arctic curve of the domain wall six-vertex model in its antiferroelectric regime. J. Stat. Mech. 2010, L03002 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Colomo, F., Pronko, A.G.: The arctic curve of the domain wall six-vertex model in its antiferroelectric regime. J. Stat. Mech. 2010, L03002 (2010)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Colomo, F., Noferini, V., Pronko, A.G.: Algebraic arctic curves in the domain-wall six-vertex model. J. Phys. A: Math. Theor. 44, 195201 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Kadanoff, L.P., Wegner, F.J.: Some critical properties of the eight-vertex model. Phys. Rev. B 4, 3989 (1971)ADSCrossRefGoogle Scholar
  40. 40.
    Wu, F.Y.: Ising model with 4-spin interactions. Phys. Rev. B 4, 2312 (1971)ADSCrossRefGoogle Scholar
  41. 41.
    Barber, M.N.: Non-universality in the Ising-model with nearest and next-nearest neighbor interactions. J. Math. A Math. Gen. 12, 679 (1979)ADSCrossRefGoogle Scholar
  42. 42.
    Minami, K., Suzuki, M.: A 2-dimensional Ising-model with nonuniversal critical-behavior. Physica A 195, 457 (1993)ADSCrossRefGoogle Scholar
  43. 43.
    Landau, D.P.: Phase-transitions in the Ising square lattice with next-nearest-neighbor interactions. Phys. Rev. B 21, 1285 (1980)ADSCrossRefGoogle Scholar
  44. 44.
    Binder, K., Landau, D.P.: Phase-diagrams and critical-behavior in Ising square lattices with nearest-neighbor and next-nearest-neighbor Interactions. Phys. Rev. B 21, 1941 (1980)ADSCrossRefGoogle Scholar
  45. 45.
    Landau, D.P., Binder, K.: Phase-diagrams and critical-behavior of Ising square lattices with nearest-neighbor, next-nearest-neighbor, and 3rd-nearest-neighbor couplings. Phys. Rev. B 31, 5946 (1985)ADSCrossRefGoogle Scholar
  46. 46.
    Buzano, C., Pretti, M.: Cluster variation approach to the Ising square lattice with two- and four-spin interactions. Phys. Rev. B 56, 636 (1997)ADSCrossRefGoogle Scholar
  47. 47.
    Cirillo, E., Gonnella, G., Troccoli, M., Maritan, A.: Correlation functions by cluster variation method for Ising model with NN, NNN, and plaquette interactions. J. Stat. Phys. 94, 67 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Jack, R.L., Garrahan, J.P., Sherrington, D.S.: Glassy behavior in an exactly solved spin system with a ferromagnetic transition. Phys. Rev. E 71, 036112 (2005)ADSCrossRefGoogle Scholar
  49. 49.
    Jack, R.L., Berthier, L., Garrahan, J.P.: Static and dynamic length scales in a simple glassy plaquette model. Phys. Rev. E 72, 016103 (2005)ADSCrossRefGoogle Scholar
  50. 50.
    Jack, R.L., Berthier, L., Garrahan, J.P.: Fluctuation-dissipation relations in plaquette spin systems with multi-stage relaxation. J. Stat. Mech. 2006, P12005 (2006)CrossRefGoogle Scholar
  51. 51.
    Turner, R.M., Jack, R.L., Garrahan, J.P.: Overlap and activity glass transitions in plaquette spin models with hierarchical dynamics. Phys. Rev. E 92, 022115 (2015)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    Jack, R.L., Garrahan, J.P.: Phase transition for quenched coupled replicas in a plaquette spin model of glasses. Phys. Rev. Lett. 116, 055702 (2016)ADSCrossRefGoogle Scholar
  53. 53.
    Heyderman, L.J., Stamps, R.L.: Artificial ferroic systems: novel functionality from structure, interactions and dynamics. J. Phys. Cond. Matt. 25, 363201 (2013)CrossRefGoogle Scholar
  54. 54.
    Nisoli, C., Moessner, R., Schiffer, P.: Artificial spin ice: designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473 (2013)ADSCrossRefGoogle Scholar
  55. 55.
    Wang, R.F., et al.: Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006)ADSCrossRefGoogle Scholar
  56. 56.
    Nisoli, C., et al.: Ground state lost but degeneracy found: the effective thermodynamics of artificial spin ice. Phys. Rev. Lett. 98, 217203 (2007)ADSCrossRefGoogle Scholar
  57. 57.
    Morgan, J.P., Stein, A., Langridge, S., Marrows, C.H.: Thermal ground-state ordering and elementary excitations in artificial magnetic square ice. Nat. Phys. 7, 75 (2011)CrossRefGoogle Scholar
  58. 58.
    Morgan, J.P., et al.: Real and effective thermal equilibrium in artificial square spin ices. Phys. Rev. B 87, 024405 (2013)ADSCrossRefGoogle Scholar
  59. 59.
    Phatak, C., Petford-Long, A.K., Heinonen, O., Tanase, M., De Graef, M.: Nanoscale structure of the magnetic induction at monopole defects in artificial spin-ice lattices. Phys. Rev. B 83, 174431 (2011)ADSCrossRefGoogle Scholar
  60. 60.
    Remhof, A., et al.: Magnetostatic interactions on a square lattice. Phys. Rev. B 77, 134409 (2008)ADSCrossRefGoogle Scholar
  61. 61.
    Mengotti, E., et al.: Building blocks of an artificial kagome spin ice: photoemission electron microscopy of arrays of ferromagnetic islands. Phys. Rev. B 78, 144402 (2008)ADSCrossRefGoogle Scholar
  62. 62.
    Ladak, S., Read, D., Perkins, G., Cohen, L., Branford, W.: Direct observation of magnetic monopole defects in an artificial spin-ice system. Nat. Phys. 6, 359 (2010)CrossRefGoogle Scholar
  63. 63.
    Li, J., et al.: Comparing artificial frustrated magnets by tuning the symmetry of nanoscale permalloy arrays. Phys. Rev. B 81, 092406 (2010)ADSCrossRefGoogle Scholar
  64. 64.
    Li, J., et al.: Comparing frustrated and unfrustrated clusters of single-domain ferromagnetic islands. Phys. Rev. B 82, 134407 (2010)ADSCrossRefGoogle Scholar
  65. 65.
    Farhan, A., et al.: Direct observation of thermal relaxation in artificial spin ice. Phys. Rev. Lett. 111, 057204 (2013)ADSCrossRefGoogle Scholar
  66. 66.
    Nisoli, C., et al.: Effective temperature in an interacting vertex system: theory and experiment on artificial spin ice. Phys. Rev. Lett. 105, 1 (2010)CrossRefGoogle Scholar
  67. 67.
    Möller, G., Moessner, R.: Artificial square ice and related dipolar nanoarrays. Phys. Rev. Lett. 96, 1–4 (2006)CrossRefGoogle Scholar
  68. 68.
    Wysin, G.M., Moura-Melo, W.A., Mól, L.A.S., Pereira, A.R.: Dynamics and hysteresis in square lattice artificial spin ice. New J. Phys. 15, 045029 (2013)ADSCrossRefGoogle Scholar
  69. 69.
    Bethe, H.A.: Statistical theory of superlattices. Proc. R. Soc. Lond. A 150, 552 (1935)ADSCrossRefMATHGoogle Scholar
  70. 70.
    Pearl, J.: Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Francisco (1988)MATHGoogle Scholar
  71. 71.
    Yedidia, J.S., Freeman, W.T., Weiss, Y.: Understanding belief propagation and its generalizations. In: Lakemeyer, G., Nebel, B. (eds.) Exploring Artificial Intelligence in the New Millennium, pp. 239–269. Morgan Kaufmann Publishers Inc, San Francisco (2003)Google Scholar
  72. 72.
    Pelizzola, A.: Cluster variation method in statistical physics and probabilistic graphical models. J. Phys. A Math. Gen. 38, R309–R339 (2005)ADSMathSciNetCrossRefGoogle Scholar
  73. 73.
    Mézard, M., Montanari, A.: Information, Physics, and Computation (Oxford Graduate Texts). Oxford University Press, Oxford (2009)CrossRefMATHGoogle Scholar
  74. 74.
    Cirillo, E., Gonnella, G., Pelizzola, A.: Folding transitions of the triangular lattice with defects. Phys. Rev. E 53, 1479 (1996)ADSCrossRefMATHGoogle Scholar
  75. 75.
    Cirillo, E., Gonnella, G., Pelizzola, A.: Folding transitions in three-dimensional space with defects. Nucl. Phys. B 862, 821–834 (2012)ADSCrossRefMATHGoogle Scholar
  76. 76.
    Budrikis, Z., et al.: Domain dynamics and fluctuations in artificial square ice at finite temperatures. New J. Phys. 14, 035014 (2012)ADSCrossRefGoogle Scholar
  77. 77.
    Chalker, J.T.: Geometrically Frustrated Antiferromagnets: Statistical Mechanics and Dynamics. Springer, Dordrecht (2011)Google Scholar
  78. 78.
    Villain, J., Bidaux, R., Carton, J.-P., Conte, R.: Order as an effect of disorder. J. Phys. 41, 1263 (1980)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Evertz, H.G., Lana, G., Marcu, M.: Cluster algorith for vertex models. Phys. Rev. Lett. 70, 875 (1993)ADSCrossRefGoogle Scholar
  80. 80.
    Lammert, P.E., Crespi, V.H., Nisoli, C.: Gibbsianizing nonequilibrium dynamics of artificial spin ice and other spin systems. New J. Phys. 14, 045009 (2012)ADSCrossRefGoogle Scholar
  81. 81.
    Castelnovo, C., Moessner, R., Sondhi, S.L.: Magnetic monopoles in spin ice. Nature 451, 42–45 (2008)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique et Hautes EnergiesSorbonne Universités, Université Pierre et Marie CurieParisFrance

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