Skip to main content
SpringerLink
Log in

Quantum Spin Liquids and Fractionalization

  • Chapter
  • First Online:
Introduction to Frustrated Magnetism

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 164))

Abstract

This chapter discusses quantum antiferromagnets which do not break any symmetries at zero temperature – also called “spin liquids” – and focuses on lattice spin models with Heisenberg-like (i.e. SU(2)-symmetric) interactions in dimensions larger than one. We begin by discussing the Lieb–Schultz–Mattis theorem and its recent extension to D > 1 by Hastings (2004), which establishes an important distinction between spin liquids with an integer and with a half-integer spin per unit cell. Spin liquids of the first kind, “band insulators”, can often be understood by elementary means, whereas the latter, “Mott insulators”, are more complex (featuring “topological order”) and support spin-1/2 excitations (spinons). The fermionic formalism (Affleck and Marston, 1988) is described and the effect of fluctuations about mean-field solutions, such as the possible creation of instabilities, is discussed in a qualitative way. In particular, we explain the emergence of gauge modes and their relation to fractionalization. The concept of the projective symmetry group (X.-G. Wen, 2002) is introduced, with the aid of some examples. Finally, we present the phenomenology of (gapped) short-ranged resonating-valence-bond spin liquids, and make contact with the fermionic approach by discussing their description in terms of a fluctuating Z 2 gauge field. Some recent references are given to other types of spin liquid, including gapless ones.

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 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P.W. Anderson, Mat. Res. Bull. 8, 153 (1973)

  2. X.-G. Wen, Phys. Rev. B 40, 7387 (1989)

    Article  ADS  Google Scholar 

  3. X.-G. Wen, Q. Niu, Phys. Rev. B 41, 9377 (1990)

    Article  ADS  Google Scholar 

  4. X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

    Article  ADS  Google Scholar 

  5. X.-G. Wen, Quantum Field Theory of Many-body Systems from the Origin of Sound to an Origin of Light and Electrons (Oxford University Press, Oxford, 2004)

    Google Scholar 

  6. E.H. Lieb, T.D. Schultz, D.C. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961)

    Google Scholar 

  7. M.B. Hastings, Phys. Rev. B 69, 104431 (2004)

    Article  ADS  Google Scholar 

  8. B. Nachtergaele, R. Sims, Commun. Math. Phys. 276, 437 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. P.W. Anderson, Science 235, 1196 (1987)

    Article  ADS  Google Scholar 

  10. G. Baskaran, Z. Zou, P.W. Anderson, Solid State Commun. 88, 853 (1987)

    Article  Google Scholar 

  11. I. Affleck et al., Phys. Rev. B 38, 745 (1988)

    Article  ADS  Google Scholar 

  12. X.-G. Wen, P.A. Lee, Phys. Rev. Lett. 76, 503 (1996)

    Article  ADS  Google Scholar 

  13. A.F. Andreev, I.A. Grishchuk, Sov. Phys. JETP, 60, 267 (1984)

    Google Scholar 

  14. A. Läuchli et al., Phys. Rev. Lett. 95, 137206 (2005)

    Article  ADS  Google Scholar 

  15. A. Läuchli, F. Mila, K. Penc, Phys. Rev. Lett. 97, 087205 (2006)

    Article  ADS  Google Scholar 

  16. N. Shannon, T. Momoi, P. Sindzingre, Phys. Rev. Lett. 96, 027213 (2006)

    Article  ADS  Google Scholar 

  17. C. Lhuillier, G. Misguich, in Quantum Spin Liquids and Fractionalization, ed. by Lacroix, Introduction to Frustrated Magnetism (Springer, Heidelberg, 2010)

    Google Scholar 

  18. I. Affleck et al., Phys. Rev. Lett. 59, 799 (1987).

    Article  ADS  Google Scholar 

  19. C.K. Majumdar, D.K. Ghosh, J. Math. Phys. 10, 1399 (1969)

    Article  MathSciNet  ADS  Google Scholar 

  20. P. Jordan, E. Wigner, Z. Phys. 47, 631 (1928)

    Article  ADS  Google Scholar 

  21. P.A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

    Article  ADS  Google Scholar 

  22. S. Furukawa, G. Misguich, M. Oshikawa, Phys. Rev. Lett. 96, 047211 (2006)

    Article  ADS  Google Scholar 

  23. M. Oshikawa, T. Senthil, Phys. Rev. Lett. 96, 060601 (2006)

    Article  ADS  Google Scholar 

  24. M. Oshikawa, Phys. Rev. Lett. 84, 1535 (2000)

    Article  ADS  Google Scholar 

  25. I. Affleck, Phys. Rev. B 37, 5186 (1988)

    Article  ADS  Google Scholar 

  26. I. Affleck, Phys. Rev. Lett. 54, 966 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  27. I. Affleck, J.B. Marston, Phys. Rev. B 37, 3774 (1988)

    Article  ADS  Google Scholar 

  28. J.B. Marston, I. Affleck, Phys. Rev. B 39, 11538 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  29. D. Arovas, A. Auerbach, Phys. Rev. B 38, 316 (1988)

    Article  ADS  Google Scholar 

  30. A. Auerbach, D.P. Arovas, Phys. Rev. Lett. 61, 617 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  31. N. Read, S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

    Article  ADS  Google Scholar 

  32. N. Read, S. Sachdev, Phys. Rev. B 42, 4568 (1990)

    Article  ADS  Google Scholar 

  33. N. Read, S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

    Article  ADS  Google Scholar 

  34. H.A. Ceccatto et al., Phys. Rev. B 47, 12329 (1993)

    Article  ADS  Google Scholar 

  35. For a review see F. Becca et al., in Quantum Spin Liquids and Fractionalization, ed. by Lacroix, Introduction to Frustrated Magnetism (Springer, Heidelberg, 2010)

    Google Scholar 

  36. P. Horsch, T.A. Kaplan, J. Phys. C: Solid State Phys. 16, 1203 (1983)

    Article  ADS  Google Scholar 

  37. C. Gros, R. Joynt, T.M. Rice, Phys. Rev. B 36, 381 (1987)

    Article  ADS  Google Scholar 

  38. G. Kotliar, J. Liu, Phys. Rev. B 38, 5142 (1988)

    Article  ADS  Google Scholar 

  39. X.-G. Wen, Phys. Rev. B 65, 165113 (2002)

    Article  ADS  Google Scholar 

  40. M. Hermele et al., Phys. Rev. B 70, 214437 (2004)

    Article  ADS  Google Scholar 

  41. D.S. Rokhsar, Phys. Rev. B 42, 2526 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  42. J.B. Marston, C. Zeng, J. Appl. Phys. 69, 5962 (1991)

    Article  ADS  Google Scholar 

  43. M.B. Hastings, Phys. Rev. B 63, 014413 (2000)

    Article  ADS  Google Scholar 

  44. Y. Ran et al., Phys. Rev. Lett. 98, 117205 (2007)

    Article  ADS  Google Scholar 

  45. N. Read, S. Sachdev, Nucl. Phys. B, 316, 609 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  46. For a review of quantum dimer models see R. Moessner, in Quantum Spin Liquids and Fractionalization, ed. by Lacroix, Introduction to Frustrated Magnetism (Springer, Heidelberg, 2010)

    Google Scholar 

  47. X.-G. Wen, F. Wilczek, A. Zee, Phys. Rev. B 39, 11413 (1989)

    Article  ADS  Google Scholar 

  48. G. Baskaran, P.W. Anderson, Phys. Rev. B 37, 580 (1988)

    Article  ADS  Google Scholar 

  49. M. Hermele et al., Phys. Rev. B 77, 224413 (2008)

    Article  ADS  Google Scholar 

  50. S. Kivelson, Phys. Rev. B 39, 259 (1989)

    Article  ADS  Google Scholar 

  51. N. Read, B. Chakraborty, Phys. Rev. B 40, 7133 (1989)

    Article  ADS  Google Scholar 

  52. T. Senthil, M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000)

    Article  ADS  Google Scholar 

  53. S. Liang, B. Douçot, P.W. Anderson, Phys. Rev. Lett. 61, 365 (1988)

    Article  ADS  Google Scholar 

  54. N.E. Bonesteel, Phys. Rev. B 40, 8954 (1989)

    Article  ADS  Google Scholar 

  55. S. Furukawa, G. Misguich, M. Oshikawa, J. Phys.: Condens. Matter 19, 145212 (2007)

    Google Scholar 

  56. J.B. Kogut, Rev. Mod. Phys. 51, 659 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  57. R. Moessner, S. Sondhi, Phys. Rev. Lett. 86, 1881 (2001)

    Article  ADS  Google Scholar 

  58. G. Misguich, D. Serban, V. Pasquier, Phys. Rev. Lett. 89, 137202 (2002)

    Article  ADS  Google Scholar 

  59. R. Moessner, S.L. Sondhi, E. Fradkin, Phys. Rev. B 65, 024504 (2002)

    Article  ADS  Google Scholar 

  60. For an introduction to Sp(N) antiferromagnets see Sect. 4.3 of [87]

    Google Scholar 

  61. S. Sachdev, Phys. Rev. B 45, 12377 (1992)

    Article  ADS  Google Scholar 

  62. C.H. Chung, J.B. Marston, S. Sachdev, Phys. Rev. B 64, 134407 (2001)

    Article  ADS  Google Scholar 

  63. F. Wang, A. Vishwanath, Phys. Rev. B 74, 174423 (2006)

    Article  ADS  Google Scholar 

  64. A. Kitaev, Ann. Phys. 303, 2 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  65. X.-G. Wen, Phys. Rev. Lett. 90, 016803 (2003)

    Article  ADS  Google Scholar 

  66. O.I. Motrunich, T. Senthil, Phys. Rev. Lett. 89, 277004 (2002)

    Article  ADS  Google Scholar 

  67. L. Balents, M.P.A. Fisher, S.M. Girvin, Phys. Rev. B 65, 224412 (2002)

    Article  ADS  Google Scholar 

  68. D.N. Sheng, L. Balents, Phys. Rev. Lett. 94, 146805 (2005)

    Article  ADS  Google Scholar 

  69. G. Misguich et al., Phys. Rev. B 60, 1064 (1999)

    Article  ADS  Google Scholar 

  70. J.B. Fouet, P. Sindzingre, C. Lhuillier, Eur. J. Phys. B, 20, 241 (2001)

    Article  ADS  Google Scholar 

  71. L. Capriotti et al., Phys. Rev. Lett. 87, 097201 (2001)

    Article  ADS  Google Scholar 

  72. L. Capriotti, D.J. Scalapino, S.R. White, Phys. Rev. Lett. 93, 177004 (2004)

    Article  ADS  Google Scholar 

  73. K.S. Raman, R. Moessner, S.L. Sondhi, Phys. Rev. B 74, 064413 (2005)

    Article  ADS  Google Scholar 

  74. A. Seidel, Phys. Rev. B 80, 165131 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  75. A. Kitaev, J. Preskill, Phys. Rev. Lett. 96, 110404 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  76. M. Levin, X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006)

    Article  ADS  Google Scholar 

  77. A.M. Polyakov, Nucl. Phys. B 120, 429 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  78. W. Rantner, X.-G. Wen, Phys. Rev. Lett. 86, 3871 (2001)

    Article  ADS  Google Scholar 

  79. M. Hermele, T. Senthil, M.P.A. Fisher, Phys. Rev. B 72, 104404 (2005)

    Article  ADS  Google Scholar 

  80. T. Senthil et al., Science 303, 1490 (2004)

    Article  ADS  Google Scholar 

  81. V. Kalmeyer, R.B. Laughlin, Phys. Rev. Lett. 59, 2095 (1987)

    Article  ADS  Google Scholar 

  82. M. Freedman, C. Nayak, K. Shtengel, Phys. Rev. Lett. 94, 066401 (2005)

    Article  ADS  Google Scholar 

  83. A.Y. Kitaev, Ann. Phys. 321, 2 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  84. H. Yao, S.A. Kivelson, Phys. Rev. Lett. 99, 247203 (2007)

    Article  ADS  Google Scholar 

  85. P. Fendley, arXiv:0711.0014v1

  86. J. Alicea, O.I. Motrunich, M.P.A. Fisher, Phys. Rev. Lett. 95, 241203 (2005)

    Article  Google Scholar 

  87. G. Misguich, C. Lhuillier, in Frustrated spin systems, ed. by H.T. Diep (World-Scientific, Singapore, 2005) (also on cond-mat/0310405)

Download references

Author information

Authors and Affiliations

  1. Institut de Physique Théorique, Orme des Merisiers batiment 774, Point courrier 136, CEA/DSM/IPhT, CEA/Saclay, 91191, Gif-sur-Yvette Cedex, France

    Grégoire Misguich

Authors
  1. Grégoire Misguich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Grégoire Misguich .

Editor information

Editors and Affiliations

  1. Laboratoire Louis Néel, CNRS, Grenoble CX, France

    Claudine Lacroix

  2. CNRS, Labo. Physique Solides, Université Paris-Sud XI, Orsay CX, 91405, France

    Philippe Mendels

  3. Inst. Physique Théorique, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland

    Frédéric Mila

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Misguich, G. (2011). Quantum Spin Liquids and Fractionalization. In: Lacroix, C., Mendels, P., Mila, F. (eds) Introduction to Frustrated Magnetism. Springer Series in Solid-State Sciences, vol 164. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10589-0_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10589-0_16

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10588-3

  • Online ISBN: 978-3-642-10589-0

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation