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Quantum Magnetism

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An Introduction to Quantum Spin Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 816))

Abstract

Many quantum spin systems of spin quantum number s > 1/2 or in spatial dimensions greater than 1D cannot be solved exactly. One source of this “lack of integrability” comes from the competition between different bonds, as in quantum frustration. We begin this final chapter by considering the application of approximate methods to one-dimensional models such as the spin-half J 1–J 2 model, and the spin-one Heisenberg and biquadratic models . The properties of the spin-half Heisenberg model for Archimedean lattices such as the square (unfrustrated) lattice, and the triangular and Kagomé (frustrated) lattices are listed for a variety of approximate techniques. The phenomenon of “order-from-disorder” is investigated in the context of the triangular lattice antiferromagnet in the presence of an external field. Finally, the properties of the square-lattice J 1–J 2 model and the Shastry-Sutherland model are studied. The chapter shows how the application of a range of approximate techniques, in addition to the few isolated exact results, can provide a comprehensive and compelling description of the ground-state properties of a wide range of quantum spin systems.

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References

  1. Diep, T.H.: Frustrated Spin Systems. World Scientific, Singapore (2004)

    MATH  Google Scholar 

  2. Schollwöck, U., Richter, J., Farnell, D.J.J., Bishop, R.F. (eds.): Quantum Magnetism. Lecture Notes in Physics, vol. 645. Springer, Berlin (2004)

    Google Scholar 

  3. Mattis, D.C.: The Theory of Magnetism I, Springer, Berlin (1981, 1988).

    Book  Google Scholar 

  4. Caspers, W.J.: Spin Systems. World Scientific, Singapore (1989)

    Google Scholar 

  5. Läuchli, A.M.: Quantum Magnetism and Strongly Correlated Electrons in Low Dimensions. Swiss Federal Institute of Technology, Zurich (2002)

    Google Scholar 

  6. Majumdar, C.K., Ghosh, D.K.: J. Math. Phys. 10, 1388 (1969)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  8. Tonegawa, T., Harada, I.: J. Phys. Soc. Jpn. 56, 2153 (1987)

    Article  ADS  Google Scholar 

  9. Nomura, K., Okamoto, K.: Phys. Lett. 169A, 433 (1992)

    ADS  Google Scholar 

  10. Nomura, K., Okamoto, K.: J. Phys. Soc. Jpn 62, 1123 (1993)

    Article  ADS  Google Scholar 

  11. Nomura, K., Okamoto, K.: J. Phys. A. Math. Gen. 27, 5773 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Chitra, R., Pati, S., Krishnamurthy, H.R., Sen, D., Ramasesha, S.: Phys. Rev. B 52, 6581 (1995)

    Article  ADS  Google Scholar 

  13. White, S.R., Affleck, I.: Phys. Rev. B 54, 9862 (1996)

    Article  ADS  Google Scholar 

  14. Mikeska, H.-J., Kolezhuk, A.K.: One-dimensional magnetism. In: Schollwöck, U., Richter, J., Farnell, D.J.J., Bishop, R.F. (eds.) Quantum Magnetism. Lecture Notes in Physics, vol. 645, pp. 1–83. Springer, Berlin (2004)

    Google Scholar 

  15. Aligia, A.A., Batista, C.D., Eßler, F.H.L.: Phys. Rev. B 62, 3259 (2000)

    Article  ADS  Google Scholar 

  16. Zinke, R., Drechsler, S.-L., Richter, J.: Phys. Rev. B 79, 094425 (2009)

    Article  ADS  Google Scholar 

  17. Farnell, D.J.J., Parkinson, J.B.: J. Phys. Condens. Matter 6, 5521 (1994)

    Article  ADS  Google Scholar 

  18. Bursill, R., Gehring, G.A., Farnell, D.J.J., Parkinson, J.B., Xiang, T., Zeng, C.: J. Phys. Condens. Matter 7, 8605 (1995)

    Article  ADS  Google Scholar 

  19. Farnell, D.J.J., Richter, J., Zinke, R., Bishop, R.F.: J. Stat. Phys. 135, 175 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. White, S.R., Huse, D.A.: Phys. Rev. B 48, 3844 (1993)

    Article  ADS  Google Scholar 

  21. The exact diagonalisations SpinPack code of J. Schulenburg is available under GPL licence at: http://www.ovgu.de/jschulen/spin/

  22. Farnell, D.J.J., Gernoth, K.A., Bishop, R.F.: J. Stat. Phys. 108, 401 (2002)

    Article  MATH  Google Scholar 

  23. Haldane, F.D.M.: Phys. Lett. 93A, 464–468 (1983)

    MathSciNet  ADS  Google Scholar 

  24. Haldane, F.D.M.: Phys. Rev. Lett. 50, 1153–1156 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  25. Nomura, K.: Phys. Rev. B 40, 9142 (1989)

    Article  ADS  Google Scholar 

  26. Takhtajan, L.A.: Phys. Lett. 87 A, 479–482 (1982)

    MathSciNet  ADS  Google Scholar 

  27. Babujian, H.M.: Phys. Lett. 90 A, 479–482 (1982)

    ADS  Google Scholar 

  28. Lai, C.K.: J. Math. Phys. 15, 1675–1676 (1974)

    Article  ADS  Google Scholar 

  29. Sutherland, B.: Phys. Rev. B 12, 3795–3805 (1975)

    Article  ADS  Google Scholar 

  30. Parkinson, J.B.: J. Phys. C Solid State Phys. 20, L1029–L1032 (1987)

    Article  ADS  Google Scholar 

  31. Parkinson, J.B.: J. Phys. C Solid State Phys. 21, 3793–3806 (1988)

    Article  ADS  Google Scholar 

  32. Barber, M.N., Bachelor, M.T.: Phys. Rev. B 40, 4621–4626 (1989)

    Article  ADS  Google Scholar 

  33. Klümper, A.: Europhys. Lett. 9, 815–820 (1989)

    Article  ADS  Google Scholar 

  34. Klümper, A.: J. Phys. A Math. Gen. 23, 809–823 (1990)

    Article  ADS  Google Scholar 

  35. Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Phys. Rev. Lett. 59, 799–802 (1987)

    Article  ADS  Google Scholar 

  36. Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Commun. Math. Phys. 115, 477–528 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  37. Chubukov, A.V.: J. Phys.: Condens. Matter 2, 1593–1608 (1990)

    Article  ADS  Google Scholar 

  38. Chubukov, A.V.: Phys. Rev. B 43, 3337–3344 (1991)

    Article  ADS  Google Scholar 

  39. Fáth, G., Sólyom, J.: Phys. Rev. B 44, 11836–11844 (1991)

    Article  ADS  Google Scholar 

  40. Fáth, G., Sólyom, J.: Phys. Rev. B 47, 872–881 (1994)

    Article  Google Scholar 

  41. Xiang, T., Gehring, G.: Phys. Rev. B 48, 303–310 (1993)

    Article  ADS  Google Scholar 

  42. Manousakis, E.: Rev. Mod. Phys. 63, 1 (1991)

    Article  ADS  Google Scholar 

  43. Sandvik, A.W.: Phys. Rev. B 56, 11678 (1997)

    Article  ADS  Google Scholar 

  44. Hamer, C.J., Weihong, Z., Arndt, P.: Phys. Rev. B 46, 6276 (1992)

    Article  ADS  Google Scholar 

  45. Weihong, Z., Hamer, C.J.: Phys. Rev. B 43, 8321 (1991)

    Article  ADS  Google Scholar 

  46. Zeng, W., Oitmaa, J., Hamer, C.J.: Phys. Rev. B 43, 8321 (1991)

    Article  ADS  Google Scholar 

  47. Reger, J.D., Riera, J.A., Young, A.P.: J. Phys. Condens. Matter 1, 1855 (1989)

    Article  ADS  Google Scholar 

  48. Weihong, Z., Oitmaa, J., Hamer, C.J.: Phys. Rev. B 44, 11869 (1991)

    Article  ADS  Google Scholar 

  49. Oitmaa, J., Hamer, C.J., Weihong, Z.: Phys. Rev. B 45, 9834 (1992)

    Article  ADS  Google Scholar 

  50. Farnell, D.J.J., Bishop, R.F.: Int. J. Mod. Phys. B 22, 3369 (2008)

    Article  ADS  Google Scholar 

  51. Troyer, M., Kontani, H., Ueda, K.: Phys. Rev. Lett. 76, 3822 (1996)

    Article  ADS  Google Scholar 

  52. Ueda, K., Kontani, H., Sigrist, M., Lee, P.A.: Phys. Rev. Lett. 76, 1932 (1996)

    Article  ADS  Google Scholar 

  53. Weihong, Z., Gelfand, M.P., Singh, R.R.P., Oitmaa, J., Hamer, C.J.: Phys. Rev. B 55, 11377 (1997)

    Article  ADS  Google Scholar 

  54. Farnell, D.J.J., Schulenburg, J., Richter, J., Gernoth, K.A.: Phys. Rev. B 72, 172408 (2005)

    Article  ADS  Google Scholar 

  55. Albrecht, M., Mila, F.: Europhys. Lett. 34, 145 (1996).

    Article  ADS  Google Scholar 

  56. Weihong, Z., Hamer, C.J., Oitmaa, J.: Phys. Rev. B 60, 6608 (1999).

    Article  ADS  Google Scholar 

  57. Miyake, S.J.: J. Phys. Soc. Jpn. 61, 983 (1992)

    Article  ADS  Google Scholar 

  58. Singh, R.P., Huse, D.A.: Phys. Rev. Lett. 68, 1766 (1992)

    Article  ADS  Google Scholar 

  59. Schmallfuß, D., Tomczak, P., Schulenburg, J., Richter, J.: Phys. Rev. B 63, 22405 (2002)

    Google Scholar 

  60. Harris, A.B., Kallin, C., Berlinsky, A.J.: Phys. Rev. B 45, 2899 (1993).

    Article  ADS  Google Scholar 

  61. Farnell, D.J.J., Bishop, R.F., Gernoth, K.A.: Phys. Rev. B 63, 220402(R) (2001)

    Article  ADS  Google Scholar 

  62. Collins, M.F., Petrenko, O.A.: Can. J. Phys. 75, 655 (1997)

    Article  Google Scholar 

  63. Honecker, A.: J. Phys. Condens. Matter 11, 4697 (1999)

    Article  ADS  Google Scholar 

  64. Cabra, D.C., Grynberg, M.D., Honecker, A., Pujol, P.: In: Hernández, S., Clark, J.W. (eds.) Magnetization Plateaux in Quasi-One-Dimensional Strongly Correlated Electron Systems. Condens. Matter Theor., vol. 16, p. 17. Nova Science Publishers, New York, NY (2001); arXiv:cond-mat/0010376v1

    Google Scholar 

  65. Lhuillier, C., Misguich, G.: In: Berthier, C., Lévy, L.P., Martinez, G. (eds.) Frustrated Quantum Magnets. High Magnetic Fields. Lecture Notes in Physics, vol. 595, p. 161. Springer, Berlin (2002)

    Google Scholar 

  66. Honecker, A., Schulenburg, J., Richter, J.: J. Phys. Condens. Matter 16, S749 (2004)

    Article  ADS  Google Scholar 

  67. Nishimori, H., Miyashita, S.: J. Phys. Soc. Jpn. 55, 4448 (1986)

    Article  ADS  Google Scholar 

  68. Chubukov, A.V., Golosov, D.I.: J. Phys. Condens. Matter 3, 69 (1991)

    Article  ADS  Google Scholar 

  69. Alicea, J., Chubukov, A.V., Starykh, O.A.: Phys. Rev. Lett. 102, 137201 (2009)

    Article  ADS  Google Scholar 

  70. Chubukov, A.V., Sachdev, S., Senthil, T.: J. Phys. Condens. Matter 6, 8891 (1994)

    Article  ADS  Google Scholar 

  71. Trumper, A.E., Capriotti, L., Sorella, S.: Phys. Rev. B 61, 11529 (2000)

    Article  ADS  Google Scholar 

  72. Schulenburg, J., Richter, J.: Phys. Rev. 65, 054420 (2002)

    Article  ADS  Google Scholar 

  73. Ono, T., Tanaka, H., Aruga Katori, H., Ishikawa, F., Mitamura, H., Goto, T.: Phys. Rev. B 67, 104431 (2003)

    Article  ADS  Google Scholar 

  74. Farnell, D.J.J., Richter, J., Zinke, R., J. Phys. Condens. Matter 21, 406002 (2009)

    Article  Google Scholar 

  75. Fortune, N.A., Hannahs, S.T., Yoshida, Y., Sherline, T.E., Ono, T., Tanaka, H., Takano, Y.: arXiv:0812.2077v1

    Google Scholar 

  76. Bishop, R.F., Farnell, D.J.J., Parkinson, J.B.: Phys. Rev. B 58, 6394 (1998)

    Article  ADS  Google Scholar 

  77. Bishop, R.F., Li, P.H.Y., Darradi, R., Schulenburg, J., Richter, J.: Phys. Rev. B 78, 054412 (2008)

    Article  ADS  Google Scholar 

  78. Bishop, R.F., Li, P.H.Y., Darradi, R., Richter, J.: J. Phys. Condens. Matter 20, 255251 (2008)

    Article  ADS  Google Scholar 

  79. Darradi, R., Richter, J., Schulenburg, J.: J. Phys. Conf. Ser. 145, 012049 (2009)

    Article  ADS  Google Scholar 

  80. Darradi, R., Derzhko, O., Zinke, R., Schulenburg, J., Krüger, S.E., Richter, J.: Phys. Rev. B 78, 214415 (2008)

    Article  ADS  Google Scholar 

  81. Oitmaa, J., Weihong, Z.: Phys. Rev. B 54, 3022 (1996)

    Article  ADS  Google Scholar 

  82. Singh, R.R.P., Weihong, Z., Hamer, C.J., Oitmaa, J.: Phys. Rev. B 60, 7278 (1999)

    Article  ADS  Google Scholar 

  83. Kotov, V.N., Oitmaa, J., Sushkov, O., Weihong, Z.: Philos. Mag. B 80, 1483 (2000)

    Article  ADS  Google Scholar 

  84. Sirker, J., Weihong, Z., Sushkov, O.P., Oitmaa, J.: Phys. Rev. B 73, 184420 (2006)

    Article  ADS  Google Scholar 

  85. Pardini, T., Singh, R.R.P.: Phys. Rev. B 79, 094413 (2009)

    Article  ADS  Google Scholar 

  86. Dagotto, E., Moreo, A.: Phys. Rev. Lett. 63, 2148 (1989)

    Article  ADS  Google Scholar 

  87. Dagotto, E., Moreo, A.: Phys. Rev. B 39, 4744(R) (1989)

    Article  ADS  Google Scholar 

  88. Schulz, H.J., Ziman, T.A.L.: Europhys. Lett. 18, 355 (1992)

    Article  ADS  Google Scholar 

  89. Schulz, H.J., Ziman, T.A.L., Poilblanc, D.: J. Phys. I 6, 675 (1996)

    Article  Google Scholar 

  90. Richter, J., Schulenburg, J.: arXiv:0909.3723v1

    Google Scholar 

  91. Isaev, L., Ortiz, G., Dukelsky, J.: Phys. Rev. B 79, 024409 (2009)

    Article  ADS  Google Scholar 

  92. Chandra, P., Doucot, B.: Phys. Rev. B 38, 9335 (1988)

    Article  ADS  Google Scholar 

  93. Capriotti, L., Becca, F., Parola, A., Sorella, S.: Phys. Rev. Lett. 87, 097201 (2001)

    Article  ADS  Google Scholar 

  94. Shastry, B.S., Sutherland, B.: Physica B 108, 1069 (1981)

    Article  Google Scholar 

  95. Weihong, Z., Oitmaa, J., Hamer, C.J.: Phys. Rev. B 65, 014408 (2002)

    ADS  Google Scholar 

  96. Läuchli, A., Wessel, S., Sigrist, M.: Phys. Rev. B 66, 014401 (2002)

    Article  ADS  Google Scholar 

  97. Müller-Hartmann, E., Singh, R.R.P., Knetter, C., Uhrig, G.S.: Phys. Rev. Lett. 84, 1808 (2000)

    Article  ADS  Google Scholar 

  98. Koga, A., Kawakami, N.: Phys. Rev. Lett. 84, 4461 (2000)

    Article  ADS  Google Scholar 

  99. Al Hajii, M., Guihery, N., Malrieu, J.-P., Bouquillon, B.: Eur. Phys. J. B 41, 11 (2004)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  101. Miyahara, S., Ueda, K.: J. Phys.: Condens. Matter 15, R327 (2003)

    Article  ADS  Google Scholar 

  102. Darradi, R., Richter, J., Farnell, D.J.J.: Phys. Rev. B. 72, 104425 (2005)

    Article  ADS  Google Scholar 

  103. Farnell, D.J.J., Richter, J., Zinke, R., Bishop, R.F.: J. Stat. Phys. 135, 175 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Manchester, School of Mathematics, Oxford Road, M13 9PL, Manchester, UK

    John B. Parkinson

  2. Health Methodology Research Group, University of Manchester, School of Community-Based Medicine, Jean McFarlane Building, University Place, M13 9PL, Manchester, UK

    Damian J.J. Farnell

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  1. John B. Parkinson
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  2. Damian J.J. Farnell
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Correspondence to John B. Parkinson .

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Parkinson, J.B., Farnell, D.J. (2010). Quantum Magnetism. In: An Introduction to Quantum Spin Systems. Lecture Notes in Physics, vol 816. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13290-2_11

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