Skip to main content
SpringerLink
Log in

Lieb's Spin-Reflection-Positivity Method and Its Applications to Strongly Correlated Electron Systems

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this paper, we discuss the spin-reflection-positivity method introduced by Lieb [E. H. Lieb, Phys. Rev. Lett. 62:1201 (1989)] and its applications to strongly correlated electron systems in a pedagogical manner. We emphasize the important role played by the sign rule of the ground-state wave function in studying a many-body system. To make our explanation more readable, we shall first review some well-known one-dimensional examples and recall the Lieb–Mattis theorem on the Heisenberg localized spin models. Then, after introducing the general theory of spin-reflection positivity, we show in detail how to use it to overcome the sign problem caused by the fermion characteristics of itinerant electrons in strongly correlated models. Finally, we establish several rigorous results on the Hubbard model, the periodic Anderson model and the Kondo lattice model.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. N. W. Ashcroft and N. D. Mermin, Solid State Physics(Holt, Rinehart, and Winston, New York, 1976).

    Google Scholar 

  2. D. C. Mattis, The Theory of Magnetism(Springer-Verlag, Berlin and Heidelberg, 1981).

    Google Scholar 

  3. N. F. Mott, Rev. Mod. Phys. 40:677 (1968); M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70:1039 (1998).

    Google Scholar 

  4. N. F. Mott, Metal-Insulator Transitions(Taylor & Francis, London, 1990); F. Gebhard, The Mott Metal-Insulator Transition(Springer, Berlin, 1997).

    Google Scholar 

  5. J. Hubbard, Proc. Roy. Soc. London Ser. A 276:238 (1963).

    Google Scholar 

  6. M. C. Gutzwiller, Phys. Rev. Lett. 10:59 (1963).

    Google Scholar 

  7. J. Kanamori, Prog. Theor. Phys. 30:275 (1963).

    Google Scholar 

  8. J. Hubbard, Proc. Roy. Soc. London Ser. A 281:401 (1964).

    Google Scholar 

  9. W. F. Brinkman and T. M. Rice, Phys. Rev. B 2:4302 (1970).

    Google Scholar 

  10. P. W. Anderson, Phys. Rev. 115:2 (1959). The effective Hamiltonian approach has recently been rigorously justified. See N. Datta, R. Fernandez, and J. Frölich, J. Stat. Phys. 96:545 (1999).

    Google Scholar 

  11. For a review of the Monte Carlo numerical techniques on the Hubbard model, see D. J. Scalapino, in Proceedings of the International School of Physics, Enrico Fermi Course CXXI, R. A. Borglia and J. R. Schrieffer, eds. (North-Holland, Amsterdam, 1994).

  12. H. Q. Lin and J. E. Gubernatis, Comput. Phys. 7:400 (1993); E. Dagotto, Rev. Mod. Phys. 66:763 (1994).

    Google Scholar 

  13. For a review on applications of the quantum field theories to the strongly correlated electron systems, see E. Fradkin, Field Theories of Condensed Matter Systems(Addison-Wesley, Redwood City, California, 1991); also see A. M. Tsvelik, Quantum Field Theory in Condensed Matter Physics(Cambridge University Press, Cambridge, 1995).

  14. S. R. White, Phys. Rev. Lett. 69:2863 (1992); Phys. Rev. B 48:10345 (1993); R. J. Bursill, T. Xiang, and G. A. Gehring, J. Phys. Condens. Matter 8:L583 (1996).

    Google Scholar 

  15. W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62:324 (1989).

    Google Scholar 

  16. For a review, see A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68:13 (1996).

    Google Scholar 

  17. E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20:1445 (1968).

    Google Scholar 

  18. Y. Nagaoka, Phys. Rev. 147:392 (1966). For simplified proofs of Nagaoka' theorem and its extension, see also H. Tasaki, Phys. Rev. B 40:9192 (1989); G. S. Tian, J. Phys. A: Math. Gen. 23:2231 (1990).

    Google Scholar 

  19. A. Mielke, J. Phys. A 24:3311 (1991); ibid. 25:4335 (1992); Phys. Lett. A 174:443 (1993); Phys. Rev. Lett. 82:4312 (1999); A. Mielke and H. Tasaki, Comm. Math. Phys. 158:341 (1993).

    Google Scholar 

  20. H. Tasaki, Phys. Rev. Lett. 69:1608 (1992); ibid. 73:1158 (1994); ibid. 75:4678 (1995).

    Google Scholar 

  21. E. Lieb and D. Mattis, Phys. Rev. 125:164 (1962).

    Google Scholar 

  22. G. S. Tian and H. Q. Lin, Phys. Rev. B 67:245105 (2003).

    Google Scholar 

  23. E. H. Lieb, Phys. Rev. Lett. 62:1201 (1989).

    Google Scholar 

  24. E. Lieb and D. Mattis, J. Math. Phys. 3:749 (1962).

    Google Scholar 

  25. K. Ueda, H. Tsunetsugu, and M. Sigrist, Phys. Rev. Lett. 68:1030 (1992).

    Google Scholar 

  26. T. Yanagisawa and Y. Shimoi, Phys. Rev. Lett. 74:4939 (1995); S. Q. Shen, Phys. Rev. B 53:14252 (1996); H. Tsunetsugu, Phys. Rev. B 55:3042 (1997).

    Google Scholar 

  27. J. K. Freericks and E. H. Lieb, Phys. Rev. B 51:2812 (1995).

    Google Scholar 

  28. Z. J. Wang and X. M. Qiu, Comm. Theoret. Phys. 28:51 (1997).

    Google Scholar 

  29. C. Noce, Phys. Rev. B 66:233204 (2002).

    Google Scholar 

  30. P. A. Lee, T. M. Rice, J. W. Serene, L. J. Sham, and J. W. Wilkins, Comments Condens. Matter Phys. 12:99 (1986).

    Google Scholar 

  31. G. Aeppli and Z. Fisk, Comments Condens. Matter Phys. 16:155 (1992).

    Google Scholar 

  32. For a recent review on these models, see H. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod. Phys. 69:809 (1997).

    Google Scholar 

  33. K. Kubo and T. Kishi, Phys. Rev. B 41:4866 (1990).

    Google Scholar 

  34. G. S. Tian, Phys. Rev. B 45:3145 (1992).

    Google Scholar 

  35. G. S. Tian, Phys. Rev. B 50:6246 (1994); ibid. 63:224413 (2001).

    Google Scholar 

  36. S. Q. Shen, Z. M. Qiu, and G. S. Tian, Phys. Rev. Lett. 72:1280 (1994); G. S. Tian and T. H. Lin, Phys. Rev. B 53:8196 (1996).

    Google Scholar 

  37. G. S. Tian, Phys. Rev. B 58:7612 (1998); G. S. Tian and L. H. Tang, Phys. Rev. B 60:11336 (1999).

    Google Scholar 

  38. B. S. Shastry, H. R. Krishnamurthy, and P. W. Anderson, Phys. Rev. B 41:2375 (1990); W. von der Linden and D. M. Edwards, J. Phys. Condens. Matter 3:4917 (1991); A. Barbieri, J. A. Riera, and A. P. Young, Phys. Rev. B 41:11697 (1990); G. S. Tian, Phys. Rev. B 44:4444 (1991). For a review, see T. Hanisch, G. S. Uhrig, and E. Müller-Hartmann, Phys. Rev. B 56:13960 (1997).

    Google Scholar 

  39. E. H. Lieb, _M. Loss, and _R. J. McCann, J. Math. Phys. 34:891 (1993); G. S. Tian, J. Phys. A 27:3635 (1994); Phys. Lett. A 228:383 (1997).

    Google Scholar 

  40. E. H. Lieb, in The Hubbard model, its Physics and Mathematical Physics, Nato ASI, Series B:Physics, Vol. 343, Baeriswyl, D. K. Campbell, J. M. P. Carmelo, F. Guinea, and E. Louis, eds. (Plenum, New York, 1995), and references therein.

    Google Scholar 

  41. C. N. Yang and S. C. Zhang, Mod. Phys. Lett. B 4:759 (1990).

    Google Scholar 

  42. H. Shiba, Prog. Theor. Phys. 48:2171 (1972).

    Google Scholar 

  43. E. H. Lieb and B. Nachtergaele, Phys. Rev. B 51:4777 (1995).

    Google Scholar 

  44. K. Osterwalder and R. Schrader, Comm. Math. Phys. 31:83 (1973).

    Google Scholar 

  45. J. Fröhlich, B. Simon, and T. Spencer, Comm. Math. Phys. 50:79 (1976).

    Google Scholar 

  46. F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys. 18:335 (1978).

    Google Scholar 

  47. J. Fröhlich and E. Lieb, Comm. Math. Phys. 60:233 (1978).

    Google Scholar 

  48. J. Fröhlich, R. Israel, E. Lieb, and B. Simon, Comm. Math. Phys. 62:1 (1978); J. Stat. Phys. 22:297 (1980).

    Google Scholar 

  49. L. I. Schiff, Quantum Mechanics(McGraw-Hill, New York, 1968).

    Google Scholar 

  50. N. S. Trudinger, Comm. Pure Appl. Math. 20:721 (1967).

    Google Scholar 

  51. T. Matsubara and H. Matsuda, Prog. Theor. Phys. 16:569 (1956).

    Google Scholar 

  52. W. Marshall, Proc. R. Soc. London A 232:48 (1955).

    Google Scholar 

  53. G. S. Tian, J. Phys. A 27:2305 (1994).

    Google Scholar 

  54. G. S. Tian, Phys. Rev. B 56:5355 (1997).

    Google Scholar 

  55. O. Kahn, Y. Pei, M. Verdaguer, J.-P. Renard, and J. Sletten, J. Am. Chem. Soc. 110:782 (1988); O. Kahn, Y. Pei, and Y. Journaux, in Inorganic Materials, D. W. Bruce and D. O'Hare, eds. (Wiley, New York, 1995).

    Google Scholar 

  56. A. K. Kolezhuk, H.-J. Mikeska, and S. Yamamoto, Phys. Rev. B 55:R3363 (1997); S. Brehmer, H.-J. Mikeska, and S. Yamamoto, J. Phys. Condens. Matter 9:3921 (1997).

    Google Scholar 

  57. S. K. Pati, S. Ramasesha, and D. Sen, Phys. Rev. B 55:8894 (1997); J. Phys. Condens. Matter 9:8707 (1997).

    Google Scholar 

  58. Z. J. Li, H. Q. Lin, and K. L. Yao, J. Chem. Phys. 109, 10082 (1998).

    Google Scholar 

  59. C. J. Wu, B. Chen, X. Dai, Y. Yu, and Z. B. Su, Phys. Rev. B 60:1057 (1999).

    Google Scholar 

  60. S. Yamamoto and T. Fukui, Phys. Rev. B 57:R14008 (1998); T. Sakai and S. Yamamoto, ibid. 60:4053 (1999); S. Yamamoto, T. Fukui, and T. Sakai, Eur. Phys. J. B 15:211 (2000).

    Google Scholar 

  61. N. B. Ivanov, J. Richter, and U. Schollwöck, Phys. Rev. B 58:14456 (1998); N. B. Ivanov and J. Richter, ibid. 63:144429 (2001).

    Google Scholar 

  62. C. N. Yang, Rev. Mod. Phys. 34:694 (1962).

    Google Scholar 

  63. R. Micnas, J. Ranninger, and S. Robaszkiewicz, Rev. Mod. Phys. 62:113 (1990) and references therein.

    Google Scholar 

  64. Yu. V. Korshak, T. V. Medvedera, A. A. Ovchinnikov, and V. N. Spector, Nature 326:370 (1987); K. Yoshisawa, K. Tanaka, T. Yamabe, and J. Yamauch, J. Chem. Phys. 96:5516 (1992); J. B. Torrance, S. Oostra, and A. Nazal, Synth. Met. 19:809 (1987).

    Google Scholar 

  65. Z. Fang, Z. L. Liu, and K. L. Yao, Phys. Rev. B 49:3916 (1994); ibid. 51:1304 (1995).

    Google Scholar 

  66. A. M. S. Macêdo, M. C. dos Santos, M. D. Coutinho-Filho, and C. A. Macêdo, Phys. Rev. Lett. 74:1851 (1995).

    Google Scholar 

  67. L. Calmels and A. Gold, Solid State Comm. 106:139 (1998).

    Google Scholar 

  68. E. P. Raposo and M. D. Coutinho-Filho, Phys. Rev. Lett. 78:4853 (1997); Phys. Rev. B 59:14384 (1999).

    Google Scholar 

  69. X. H. Xu, R. T. Fu, K. Hu, X. Sun, and K. Yonemitsu, Phys. Rev. B 58:9039 (1998).

    Google Scholar 

  70. G. S. Tian, J. Phys. A 30:5329 (1997); G. S. Tian and J. G. Wang, ibid. 35:941 (2002).

    Google Scholar 

  71. R. A. Horn and C. Johnson, Matrix Analysis(Cambridge University Press, Cambridge, 1985).

    Google Scholar 

  72. G. S. Tian and L. H. Tang, Phys. Rev. B 58:12333 (1998); G. S. Tian, L. H. Tang, and Q. H. Chen, Europhys. Lett. 50:361 (2000); G. S. Tian, L. H. Tang, and Q. H. Chen, Phys. Rev. B 63:054511 (2001).

    Google Scholar 

  73. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108:1175 (1957).

    Google Scholar 

  74. P. W. Anderson, J. Phys. Chem. Solids 11:28 (1959).

    Google Scholar 

  75. D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett. 74:3241 (1995); C. T. Black, D. C. Ralph, and M. Tinkham, Phys. Rev. Lett. 76:688 (1996).

    Google Scholar 

  76. J. von Delft and D. C. Ralph, Phys. Rep. 345:61 (2001).

    Google Scholar 

  77. K. A. Matveev and A. I. Larkin, Phys. Rev. Lett. 78:3749 (1997).

    Google Scholar 

  78. J. von Delft, A. D. Zaikin, D. S. Golubev, and W. Tichy, Phys. Rev. Lett. 77:3189 (1996); A. Mastellone, G. Falci, and R. Fazio, ibid. 80:4542 (1998).

    Google Scholar 

  79. C. C. Yu and S. R. White, Phys. Rev. Lett. 71:3866 (1993); T. Nishino and K. Ueda, Phys. Rev. B 47:12451 (1993); M. Vekic, J. W. Cannon, D. J. Scalapino, R. T. Scalettar, and R. L. Sugar, Phys. Rev. Lett. 74:2367 (1995); M. Guerrero and R. M. Noack, Phys. Rev. B 53:3707 (1996).

    Google Scholar 

  80. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I (Academic Press, New York, 1972), p. 295.

    Google Scholar 

  81. T. Kennedy, E. Lieb, and B. S. Shastry, Phys. Rev. Lett. 61:2582 (1988); J. Stat. Phys. 53:1019 (1988).

    Google Scholar 

  82. K. Kubo, Phys. Rev. Lett. 61:110 (1988); K. Kubo and T. Kishi, ibid. 61:2585 (1988).

    Google Scholar 

  83. N. Macris and B. Nachtergaele, J. Stat. Phys. 85:745 (1996).

    Google Scholar 

  84. Y. Hasegawa, P. Lederer, T. M. Rice, and P. B. Wiegmann, Phys. Rev. Lett. 63:907 (1989).

    Google Scholar 

  85. E. H. Lieb, Phys. Rev. Lett. 73:2158 (1994).

    Google Scholar 

  86. R. Peierls, Z. Phys. 80:763 (1933).

    Google Scholar 

  87. E. H. Lieb and M. Loss, Duke Math. J. 71:337 (1993).

    Google Scholar 

  88. S. C. Zhang, Phys. Rev. B 42:1012 (1990); G. S. Tian, J. Phys. A 26:2325 (1993); G. S. Tian, ibid. 30:841 (1997); N. Macris and C.-A. Piguet, ibid. 32:749 (1999).

    Google Scholar 

  89. C. Nayak, Phys. Rev. B 62:4880 (2000); P. J. H. Denteneer, R. T. Scalettar, and N. Trivedi, Phys. Rev. Lett. 87:146401 (2001).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Peking University, Beijing, 100871, China

    Guang-Shan Tian

Authors
  1. Guang-Shan Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tian, GS. Lieb's Spin-Reflection-Positivity Method and Its Applications to Strongly Correlated Electron Systems. Journal of Statistical Physics 116, 629–680 (2004). https://doi.org/10.1023/B:JOSS.0000037214.70064.78

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOSS.0000037214.70064.78

Search

Navigation