Skip to main content
SpringerLink
Log in

Topology of the Vacuum

  • Chapter
  • First Online:
Spin Ice

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 197))

Abstract

Before being known for the emergence of monopoles, spin ice draw the attention of the community for its extensively degenerate ground state. We have seen in previous chapters how a Coulomb gauge field emerges from the coarse-graining of this ground state. It is the goal of this chapter to connect this field-theory picture with its topological nature. In this context, spin ice is a three-dimensional vertex model, divided into topological sectors. Topological sectors are connected between each other via string updates. These strings may become the intrinsic excitations of exotic phase transitions when the degeneracy of the Coulomb phase is lifted, and can be mapped onto world lines for bosons in the corresponding quantum problem in (2+1) dimensions. As an alternative point of view, we will also discuss how the spin-ice ground state is equivalent to a fully packed loop model, whose statistics is reminiscent of critical percolation and Brownian motion in two and three dimensions respectively.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. M.J. Harris, S.T. Bramwell, D.F. McMorrow, T. Zeiske, K.W. Godfrey, Phys. Rev. Lett. 79, 2554 (1997). https://doi.org/10.1103/PhysRevLett.79.2554

  2. A.P. Ramirez, A. Hayashi, R.J. Cava, R. Siddharthan, B.S. Shastry, Nature 399, 333 (1999). https://doi.org/10.1038/20619

  3. J.C. Slater, J. Chem. Phys. 9, 16 (1941). https://doi.org/10.1063/1.1750821

  4. E.H. Lieb, Phys. Rev. 162, 162 (1967). https://doi.org/10.1103/PhysRev.162.162; Phys. Rev. Lett., 18, 1046 (1967). https://doi.org/10.1103/PhysRevLett.18.1046; Phys. Rev. Lett., 18, 692 (1967). https://doi.org/10.1103/PhysRevLett.18.692

  5. E.H. Lieb, Phys. Rev. Lett. 19, 108 (1967). https://doi.org/10.1103/PhysRevLett.19.108

  6. B. Sutherland, C.N. Yang, C.P. Yang, Phys. Rev. Lett. 19, 588 (1967). https://doi.org/10.1103/PhysRevLett.19.588

  7. B. Sutherland, Phys. Rev. Lett. 19, 103 (1967). https://doi.org/10.1103/PhysRevLett.19.103

  8. R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Dover Publications, Mineola, New-York, 2007). ISBN 10: 0486462714

    Google Scholar 

  9. R.F. Wang et al., Nature 439, 303 (2006). https://doi.org/10.1038/nature04447

  10. G. Moller, R. Moessner, Phys. Rev. Lett. 96, 237202 (2006). https://doi.org/10.1103/PhysRevLett.96.237202

  11. Y. Perrin, B. Canals, N. Rougemaille, Nature 540, 410 (2016). https://doi.org/10.1038/nature20155

  12. Erik Östman et al., Nat. Phys. 14, 375 (2018). https://doi.org/10.1038/s41567-017-0027-2

  13. S.V. Isakov, R. Moessner, S.L. Sondhi, Phys. Rev. Lett. 95, 217201 (2005). https://doi.org/10.1103/PhysRevLett.95.217201

  14. G.T. Barkema, M.E.J. Newman, Phys. Rev. E 57, 1155 (1998). https://doi.org/10.1103/PhysRevE.57.1155

  15. R.G. Melko, B.C. den Hertog, M.J.P. Gingras, Phys. Rev. Lett. 87, 067203 (2001). https://doi.org/10.1103/PhysRevLett.87.067203

  16. R.G. Melko, M.J.P. Gingras, J. Phys.: Condens. Matter 16, R1277 (2004). https://doi.org/10.1088/0953-8984/16/43/R02

  17. L.D.C. Jaubert, J.T. Chalker, P.C.W. Holdsworth, R. Moessner, Phys. Rev. Lett. 100, 067207 (2008). https://doi.org/10.1103/PhysRevLett.100.067207

  18. L.D.C. Jaubert, J.T. Chalker, P.C.W. Holdsworth, R. Moessner, Phys. Rev. Lett. 105, 087201 (2010). https://doi.org/10.1103/PhysRevLett.105.087201

  19. O.F. Syljuasen, A.W. Sandvik, Phys. Rev. E 66, 046701 (2002). https://doi.org/10.1103/PhysRevE.66.046701

  20. C.L. Henley, Annu. Rev. Condens. Matter Phys. 1, 179 (2010). https://doi.org/10.1146/annurev-conmatphys-070909-104138

  21. C. Castelnovo, C. Chamon, Phys. Rev. B 76, 174416 (2007). https://doi.org/10.1103/PhysRevB.76.174416

  22. O. Cépas, B. Canals, Phys. Rev. B 86, 024434 (2012). https://doi.org/10.1103/PhysRevB.86.024434

  23. O. Cépas, Phys. Rev. B 90, 064404 (2014). https://doi.org/10.1103/PhysRevB.90.064404

  24. C. Castelnovo, R. Moessner, S.L. Sondhi, Phys. Rev. B 84, 144435 (2011). https://doi.org/10.1103/PhysRevB.84.144435

  25. L.D.C. Jaubert, Spin 5, 1540005 (2015). https://doi.org/10.1142/S2010324715400056

  26. C. Castelnovo, R. Moessner, S.L. Sondhi, Nature 451, 42 (2008). https://doi.org/10.1038/nature06433

  27. C. Castelnovo, R. Moessner, S.L. Sondhi, Annu. Rev. Condens. Matter Phys. 3, 35 (2012). https://doi.org/10.1146/annurev-conmatphys-020911-125058

  28. A. Sen, R. Moessner, Phys. Rev. Lett. 114, 247207 (2015). https://doi.org/10.1103/PhysRevLett.114.247207

  29. L.D.C. Jaubert, P.C.W. Holdsworth, Nat. Phys. 5, 258 (2009). https://doi.org/10.1038/NPHYS1227

  30. R.A. Borzi, D. Slobinsky, S.A. Grigera, Phys. Rev. Lett. 111, 147204 (2013). https://doi.org/10.1103/PhysRevLett.111.147204

  31. I.A. Ryzhkin, J. Exp. Theor. Phys. 101, 481 (2005). https://doi.org/10.1134/1.2103216

  32. D.J.P. Morris et al., Science 326, 411 (2009). https://doi.org/10.1126/science.1178868

  33. L.D.C. Jaubert, M. Haque, R. Moessner, Phys. Rev. Lett. 107, 177202 (2011). https://doi.org/10.1103/PhysRevLett.107.177202

  34. J.L. Jacobsen, J. Kondev, Nucl. Phys. B 532, 635 (1998). https://doi.org/10.1016/S0550-3213(98)00571-9

  35. J. Kondev, Phys. Rev. Lett. 78, 4320 (1997). https://doi.org/10.1103/PhysRevLett.78.4320

  36. J.L. Jacobsen, J. Vannimenus, J. Phys. A: Math. Gen. 32, 5455 (1999). https://doi.org/10.1088/0305-4470/32/29/306

  37. H. Saleur, B. Duplantier, Phys. Rev. Lett. 58, 2325 (1987). https://doi.org/10.1103/PhysRevLett.58.2325

  38. J. Kondev, C.L. Henley, Phys. Rev. Lett. 74, 4580 (1995). https://doi.org/10.1103/PhysRevLett.74.4580

  39. A. Weinrib, B.I. Halperin, Phys. Rev. B 27, 413 (1983). https://doi.org/10.1103/PhysRevB.27.413

  40. S.V. Isakov, K. Gregor, R. Moessner, S.L. Sondhi, Phys. Rev. Lett. 93, 167204 (2004). https://doi.org/10.1103/PhysRevLett.93.167204

  41. A. Nahum, J.T. Chalker, P. Serna, M. Ortuño, A.M. Somoza, Phys. Rev. Lett. 111, 100601 (2013). https://doi.org/10.1103/PhysRevLett.111.100601

  42. B. Duplantier, H. Saleur, Phys. Rev. Lett. 59, 539 (1987). https://doi.org/10.1103/PhysRevLett.59.539

  43. D. Austin, E.J. Copeland, R.J. Rivers, Phys. Rev. D 49, 4089 (1994). https://doi.org/10.1103/PhysRevD.49.4089

  44. K.S.D. Beach, Anders W. Sandvik, Nucl. Phys. B 750, 142 (2006). https://doi.org/10.1016/j.nuclphysb.2006.05.032

  45. A. Fabricio Albuquerque, F. Alet, R. Moessner, Phys. Rev. Lett. 109, 147204 (2012). https://doi.org/10.1103/PhysRevLett.109.147204

  46. P. Fulde, K. Penc, N. Shannon, Ann. Phys. 11, 892 (2002). https://doi.org/10.1002/1521-3889(200212)11:12<892::AID-ANDP892>3.0.CO;2-J

  47. P.A. McClarty, A. O’Brien, F. Pollmann, Phys. Rev. B 89, 195123 (2014). https://doi.org/10.1103/PhysRevB.89.195123

  48. S. Kourtis, C. Castelnovo, Phys. Rev. B 94, 104401 (2016). https://doi.org/10.1103/PhysRevB.94.104401

  49. H. Ishizuka, M. Udagawa, Y. Motome, Phys. Rev. B 83, 125101 (2011). https://doi.org/10.1103/PhysRevB.83.125101

  50. L.D.C. Jaubert, S. Piatecki, M. Haque, R. Moessner, Phys. Rev. B 85, 054425 (2012). https://doi.org/10.1103/PhysRevB.85.054425

  51. J.W.F. Venderbos, M. Daghofer, J. van den Brink, S. Kumar, Phys. Rev. Lett. 109, 166405 (2012). https://doi.org/10.1103/PhysRevLett.109.166405

  52. D.A. Huse, W. Krauth, R. Moessner, S.L. Sondhi, Phys. Rev. Lett. 91, 167004 (2003). https://doi.org/10.1103/PhysRevLett.91.167004

  53. T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, M.P.A. Fisher, Science 303, 1490 (2004). https://doi.org/10.1126/science.1091806

  54. O.I. Motrunich, A. Vishwanath, Phys. Rev. B 70, 075104 (2004). https://doi.org/10.1103/PhysRevB.70.075104

  55. F. Alet et al., Phys. Rev. Lett. 94, 235702 (2005). https://doi.org/10.1103/PhysRevLett.94.235702

  56. F. Alet, G. Misguich, V. Pasquier, R. Moessner, J.L. Jacobsen, Phys. Rev. Lett. 97, 030403 (2006). https://doi.org/10.1103/PhysRevLett.97.030403

  57. G. Misguich, V. Pasquier, F. Alet, Phys. Rev. B 78, 100402 (2008). https://doi.org/10.1103/PhysRevB.78.100402

  58. D. Charrier, F. Alet, P. Pujol, Phys. Rev. Lett. 101, 167205 (2008). https://doi.org/10.1103/PhysRevLett.101.167205

  59. S. Powell, J.T. Chalker, Phys. Rev. Lett. 101, 155702 (2008). https://doi.org/10.1103/PhysRevLett.101.155702

  60. G. Chen, J. Gukelberger, S. Trebst, F. Alet, L. Balents, Phys. Rev. B 80, 045112 (2009). https://doi.org/10.1103/PhysRevB.80.045112

  61. S. Powell, J.T. Chalker, Phys. Rev. B 80, 134413 (2009). https://doi.org/10.1103/PhysRevB.80.134413

  62. P.W. Kasteleyn, J. Math. Phys. 4, 287 (1963). https://doi.org/10.1063/1.1703953

  63. G.I. Watson, J. Stat. Phys. 94, 1045 (1999). https://doi.org/10.1023/A:1004547503489

  64. K. Matsuhira, Z. Hiroi, T. Tayama, S. Takagi, T. Sakakibara, J. Phys.: Condens. Matter 14, L559 (2002). https://doi.org/10.1088/0953-8984/14/29/101

  65. M. Udagawa, M. Ogata, Z. Hiroi, J. Phys. Soc. Jpn. 71, 2365 (2002). https://doi.org/10.1143/JPSJ.71.2365

  66. R. Moessner, S.L. Sondhi, Phys. Rev. B 68, 064411 (2003). https://doi.org/10.1103/PhysRevB.68.064411

  67. T. Fennell, S.T. Bramwell, D.F. McMorrow, P. Manuel, A.R. Wildes, Nat. Phys. 3, 566 (2007). https://doi.org/10.1038/nphys632

  68. S.M. Bhattacharjee, Europhys. Lett. 15, 815 (1991). https://doi.org/10.1209/0295-5075/15/8/002

  69. J.F. Nagle, Proc. Natl. Acad. Sci. U. S. A. 70, 3443 (1973). https://doi.org/10.1073/pnas.70.12.3443

  70. Y.-Z. Chou, Y.-J. Kao, Phys. Rev. B 82, 132403 (2010). https://doi.org/10.1103/PhysRevB.82.132403

  71. S. Powell, Phys. Rev. B 91, 094431 (2015). https://doi.org/10.1103/PhysRevB.91.094431

  72. S. Powell, J.T. Chalker, Phys. Rev. B 78, 024422 (2008). https://doi.org/10.1103/PhysRevB.78.024422

  73. S. Powell, Phys. Rev. Lett. 109, 065701 (2012). https://doi.org/10.1103/PhysRevLett.109.065701

  74. S. Powell, Phys. Rev. B 87, 064414 (2013). https://doi.org/10.1103/PhysRevB.87.064414

  75. T. Fennell et al., Phys. Rev. B 72, 224411 (2005). https://doi.org/10.1103/PhysRevB.72.224411

  76. L.D.C. Jaubert, J.T. Chalker, P.C.W. Holdsworth, R. Moessner, J. Phys.: Conf. Ser. 145, 012024 (2009). https://doi.org/10.1088/1742-6596/145/1/012024

  77. H. Fukazawa, R.G. Melko, R. Higashinaka, Y. Maeno, M.J.P. Gingras, Phys. Rev. B 65, 054410 (2002). https://doi.org/10.1103/PhysRevB.65.054410

  78. S.-C. Lin, Y.-J. Kao, Phys. Rev. B 88, 220402 (2013). https://doi.org/10.1103/PhysRevB.88.220402

  79. M.L. Baez, R.A. Borzi, J. Phys.: Condens. Matter 29, 055806 (2017). https://doi.org/10.1088/1361-648X/aa4e6a

  80. S. Powell, J. Phys. A: Math. Theor. 50, 124001 (2017). https://doi.org/10.1088/1751-8121/aa5bc6

  81. J.P.C. Ruff, R.G. Melko, M.J.P. Gingras, Phys. Rev. Lett. 95, 097202 (2005). https://doi.org/10.1103/PhysRevLett.95.097202

  82. T. Yavors’kii, T. Fennell, M.J.P. Gingras, S.T. Bramwell, Phys. Rev. Lett. 101, 037204 (2008). https://doi.org/10.1103/PhysRevLett.101.037204

  83. P. Henelius et al., Phys. Rev. B 93, 024402 (2016). https://doi.org/10.1103/PhysRevB.93.024402

  84. H.R. Molavian, M.J.P. Gingras, B. Canals, Phys. Rev. Lett. 98, 157204 (2007). https://doi.org/10.1103/PhysRevLett.98.157204

  85. G.-W. Chern, R. Moessner, O. Tchernyshyov, Phys. Rev. B 78, 144418 (2008). https://doi.org/10.1103/PhysRevB.78.144418

  86. P.A. McClarty et al., Phys. Rev. B 92, 144418 (2015). https://doi.org/10.1103/PhysRevB.92.144418

  87. J.G. Rau, M.J.P. Gingras, Phys. Rev. B 92, 144417 (2015). https://doi.org/10.1103/PhysRevB.97.144417

  88. M. Mito et al., J. Magn. Magn. Mater. 310, E432 (2007). https://doi.org/10.1016/j.jmmm.2006.10.441

  89. H. Takahashi, Proc. Phys. Math. Soc. Japan 23, 548 (1941). https://doi.org/10.11429/ppmsj1919.23.0_548

  90. L. Bovo et al., Nat. Commun. 5, 3439 (2014). https://doi.org/10.1038/ncomms4439

  91. J.F. Nagle, Commun. Math. Phys. 13, 62 (1969). https://doi.org/10.1007/BF01645270

  92. L. Benguigui, Phys. Rev. B 16, 1266 (1977). https://doi.org/10.1103/PhysRevB.16.1266

  93. K.A. Ross, L. Savary, B.D. Gaulin, L. Balents, Phys. Rev. X 1, 021002 (2011). https://doi.org/PhysRevX.1.021002

  94. L. Savary, L. Balents, Phys. Rev. Lett. 108, 037202 (2012). https://doi.org/10.1103/PhysRevLett.108.037202

  95. Claudio Castelnovo, in Topological Aspects of Condensed Matter Physics, ed. by C. Chamon, M.O. Goerbig, R. Moessner, L.F. Cugliandolo. Lecture Notes of the Les Houches Summer School: Volume 103, (Oxford University Press, 2017). https://doi.org/10.1093/acprof:oso/9780198785781.003.0012

  96. D.P. Leusink et al., Appl. Phys. Lett. Mater. 2, 032101 (2014). https://doi.org/10.1063/1.4867222

  97. K. Kukli et al., Thin Solid Films 565, 261 (2014). https://doi.org/10.1016/j.tsf.2014.06.028

  98. L. Bovo, C.M. Rouleau, D. Prabhakaran, S.T. Bramwell, Nanotechnology 28, 055708 (2017). https://doi.org/10.1088/1361-6528/aa5112

  99. L.D.C. Jaubert, T. Lin, T.S. Opel, P.C.W. Holdsworth, M.J.P. Gingras, Phys. Rev. Lett. 118, 207206 (2017). https://doi.org/10.1103/PhysRevLett.118.207206

  100. E. Lantagne-Hurtubise, J.G. Rau, M.J.P. Gingras, Phys. Rev. X 8, 021053 (2018). https://doi.org/10.1103/PhysRevX.8.021053

  101. D.L. Bergman, G.A. Fiete, L. Balents, Phys. Rev. B 73, 134402 (2006). https://doi.org/10.1103/PhysRevB.73.134402

  102. O. Sikora, F. Pollmann, N. Shannon, K. Penc, P. Fulde, Phys. Rev. Lett. 103, 247001 (2009). https://doi.org/10.1103/PhysRevLett.103.247001

  103. O. Sikora, N. Shannon, F. Pollmann, K. Penc, P. Fulde, Phys. Rev. B 84, 115129 (2011). https://doi.org/10.1103/PhysRevB.84.115129

  104. K. Penc, N. Shannon, H. Shiba, Phys. Rev. Lett. 93, 197203 (2004). https://doi.org/10.1103/PhysRevLett.93.197203

  105. H. Ueda, H.A. Katori, H. Mitamura, T. Goto, H. Takagi, Phys. Rev. Lett. 94, 047202 (2005). https://doi.org/10.1103/PhysRevLett.94.047202

  106. D.L. Bergman, R. Shindou, G.A. Fiete, L. Balents, Phys. Rev. Lett. 96, 097207 (2006). https://doi.org/10.1103/PhysRevLett.96.097207

  107. M.E. Brooks-Bartlett, S.T. Banks, L.D.C. Jaubert, A. Harman-Clarke, P.C.W. Holdsworth, Phys. Rev. X 4, 011007 (2014). https://doi.org/10.1103/PhysRevX.4.011007

  108. S. Petit et al., Nat. Phys. 12, 746 (2016). https://doi.org/10.1038/nphys3710

  109. J.A.M. Paddison et al., Nat. Commun. 7, 13842 (2016). https://doi.org/10.1038/ncomms13842

  110. E. Lefrançois et al., Nat. Commun. 8, 209 (2017). https://doi.org/10.1038/s41467-017-00277-1

Download references

Acknowledgements

I am especially grateful to Peter Holdsworth, Roderich Moessner and John Chalker, with whom many of the ideas discussed in this chapter have been developed. I also acknowledge insightful discussions with Claudio Castelnovo, Adam Nahum and Stephen Powell on these topics. I would like to thank the people in the Theory of Quantum Matter Unit in Okinawa where this book was initiated, and my colleagues at the “Laboratoire d’Ondes et Matière d’Aquitaine” in Bordeaux where it was concluded.

Author information

Authors and Affiliations

  1. CNRS, LOMA, University of Bordeaux, Bordeaux, France

    L. D. C. Jaubert

Authors
  1. L. D. C. Jaubert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. D. C. Jaubert .

Editor information

Editors and Affiliations

  1. Department of Physics, Gakushuin University, Tokyo, Japan

    Masafumi Udagawa

  2. Laboratoire Ondes et Matière d’Aquitaine, Université de Bordeaux, CNRS, Bordeaux, France

    Ludovic Jaubert

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Jaubert, L.D.C. (2021). Topology of the Vacuum. In: Udagawa, M., Jaubert, L. (eds) Spin Ice. Springer Series in Solid-State Sciences, vol 197. Springer, Cham. https://doi.org/10.1007/978-3-030-70860-3_6

Download citation

Publish with us

Policies and ethics

Search

Navigation