- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Reviews of Modern Physics

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Colloquium: Exactly solvable Richardson-Gaudin models for many-body quantum systems

J. Dukelsky*, S. Pittel, and G. Sierra

J. Dukelsky*

  • Instituto de Estructura de la Materia, CSIC, Madrid, Spain

S. Pittel

  • Bartol Research Institute, University of Delaware, Newark, Delaware 19716, USA

G. Sierra

  • Instituto de Física Teórica, CSIC/UAM, Madrid, Spain

  • *Electronic address: dukelsky@iem.cfmac.csic.es
PDF

Rev. Mod. Phys. 76, 643 – Published 6 August, 2004

DOI: https://doi.org/10.1103/RevModPhys.76.643

Abstract

The use of exactly solvable Richardson-Gaudin models to describe the physics of systems with strong pair correlations is reviewed. The article begins with a brief discussion of Richardson’s early work, which demonstrated the exact solvability of the pure pairing model, and then shows how that work has evolved recently into a much richer class of exactly solvable models. The Richardson solution leads naturally to an exact analogy between these quantum models and classical electrostatic problems in two dimensions. This analogy is then used to demonstrate formally how BCS theory emerges as the large-N limit of the pure pairing Hamiltonian. Several applications to problems of relevance to condensed-matter physics, nuclear physics, and the physics of confined systems are considered. Some of the interesting effects that are discussed in the context of these exactly solvable models include (i) the crossover from superconductivity to a fluctuation-dominated regime in small metallic grains; (ii) the role of the nucleon Pauli principle in suppressing the effects of high-spin bosons in interacting boson models of nuclei, and (iii) the possibility of fragmentation in confined boson systems. Interesting insight is also provided into the origin of the superconducting phase transition both in two-dimensional electronic systems and in atomic nuclei, based on the electrostatic image of the corresponding exactly solvable quantum pairing models.

View figure in article

Article Text

References (80)

  1. Amico, L., A. Di Lorenzo, A. Mastellone, A. Osterloh, and R. Raimondi, 2002, Ann. Phys. (N.Y.) 299, 228.
  2. Amico, L., A. Di Lorenzo, and A. Osterloh, 2001, Phys. Rev. Lett. 86, 5759.
  3. Amico, L., G. Falci, and R. Fazio, 2001, J. Phys. A 34, 6425.
  4. Anderson, P. W., 1958, Phys. Rev. 112, 1900.
  5. Anderson, P. W., 1959, J. Phys. Chem. Solids 11, 28.
  6. Arias, J. E., J. Dukelsky, and J. E. García-Ramos, 2003, Phys. Rev. Lett. 91, 162502.
  7. Arrachea, L., and M. J. Rozenberg, 2001, Phys. Rev. Lett. 86, 5172, and private communication.
  8. Asorey, M., F. Falceto, and G. Sierra, 2002, Nucl. Phys. B 622, 593.
  9. Baldo, M., J. Cugnon, A. Lejeune, and U. Lombardo, 1990, Nucl. Phys. A 515, 409.
  10. Bang, J., and J. Krumlimde, 1970, Nucl. Phys. A 141, 18.
  11. Bardeen, J., L. N. Cooper, and J. R. Schrieffer, 1957, Phys. Rev. 108, 1175.
  12. Berger, S. D., and B. I. Halperin, 1998, Phys. Rev. B 58, 5213.
  13. Bethe, H. A., 1931, Z. Phys. 71, 265.
  14. Braun, F., and J. von Delft, 1998, Phys. Rev. Lett. 81, 4712.
  15. Calogero, F., 1962, J. Math. Phys. 10, 2191.
  16. Cambiaggio, M. C., A. M. F. Rivas, and M. Saraceno, 1997, Nucl. Phys. A 624, 157.
  17. Dean, D. J., and M. Hjorth-Jensen, 2003, Rev. Mod. Phys. 75, 607.
  18. Dietrich, K., H. J. Mang, and J. H. Pradal, 1964, Phys. Rev. 135, B22.
  19. Dukelsky, J., G. G. Dussal, C. Esebbag, and S. Pittel, 2004, Phys. Rev. Lett. (to be published).
  20. Dukelsky, J., C. Esebbag, and S. Pittel, 2002, Phys. Rev. Lett. 88, 062501.
  21. Dukelsky, J., C. Esebbag, and P. Schuck, 2001, Phys. Rev. Lett. 87, 066403.
  22. Dukelsky, J., and S. Pittel, 2001, Phys. Rev. Lett. 86, 4791.
  23. Dukelsky, J., J. M. Román, and G. Sierra, 2003, Phys. Rev. Lett. 90, 249803.
  24. Dukelsky, J., and P. Schuck, 2001, Phys. Rev. Lett. 86, 4207.
  25. Dukelsky, J., and G. Sierra, 1999, Phys. Rev. Lett. 83, 172.
  26. Dukelsky, J., and G. Sierra, 2000, Phys. Rev. B 61, 12302.
  27. Elliott, J. P., 1958a, Proc. R. Soc. London, Ser. A 242, 128.
  28. Elliott, J. P., 1958b, Proc. R. Soc. London, Ser. A 242, 562.
  29. Gaudin, M., 1976, J. Phys. (Paris) 37, 1087.
  30. Gaudin, M., 1995, États Propres et Valeurs Propres de l’Hamiltonien d’Appariement (Les Éditions de Physique, France).
  31. Gould, M. D., Y.-Z. Zhang, and S.-Y. Zhao, 2002, Nucl. Phys. B 630, 492.
  32. Guan, X.-W., A. Foerster, J. Links, and H.-Q. Zhou, 2002, Nucl. Phys. B 642, 501.
  33. Ha, Z. N. C., 1996, Quantum Many-Body Systems in One Dimension (World Scientific, Singapore).
  34. Haldane, F. D. M., 1981, J. Phys. C 14, 2585.
  35. Haldane, F. D. M., 1988, Phys. Rev. Lett. 60, 635.
  36. Hasegawa, M., and S. Tasaki, 1987, Phys. Rev. C 35, 1508.
  37. Hasegawa, M., and S. Tasaki, 1993, Phys. Rev. C 47, 188.
  38. Heiselberg, H., 2003, Phys. Rev. A 68, 053616.
  39. Heritier, M., 2001, Nature (London) 414, 31.
  40. Iachello, F., and A. Arima, 1980, The Interacting Boson Model (Cambridge University, Cambridge, England).
  41. Links, J., H.-Q. Zhou, M. D. Gould, and R. H. McKenzie, 2002, J. Phys. A 35, 6459.
  42. Links, J., H.-Q. Zhou, R. H. McKenzie, and M. D. Gould, 2003, J. Phys. A 36, R63.
  43. Luttinger, J. M., 1963, J. Math. Phys. 15, 609.
  44. Mastellone, A., G. Falci, and R. Fazio, 1998, Phys. Rev. Lett. 80, 4542.
  45. Matveev, K. A., and A. I. Larkin, 1997, Phys. Rev. Lett. 78, 3749.
  46. Nozières, P., and D. Saint James, 1982, J. Phys. (Paris) 43, 1133.
  47. Pan, F., and J. P. Draayer, 2002, Phys. Rev. C 66, 044314.
  48. Pittel, S., and J. Dukelsky, 2003, nucl-th/0309020.
  49. Ralph, D. C., C. T. Black, and M. Tinkham, 1997, Phys. Rev. Lett. 78, 4087.
  50. Relano, A., J. Dukelsky, J. M. G. Gomez, and J. Retamosa, 2004, Phys. Rev. E (to be published).
  51. Richardson, R. W., 1963a, Phys. Lett. 3, 277.
  52. Richardson, R. W., 1963b, Phys. Lett. 5, 82.
  53. Richardson, R. W., 1965, J. Math. Phys. 6, 1034.
  54. Richardson, R. W., 1966a, Phys. Rev. 141, 949.
  55. Richardson, R. W., 1966b, Phys. Rev. 144, 874.
  56. Richardson, R. W., 1967, Phys. Rev. 159, 792.
  57. Richardson, R. W., 1968, J. Math. Phys. 9, 1327.
  58. Richardson, R. W., 1977, J. Math. Phys. 18, 1802.
  59. Richardson, R. W., and N. Sherman, 1964, Nucl. Phys. 52, 221.
  60. Roman, J. M., G. Sierra, and J. Dukelsky, 2002, Nucl. Phys. B 634, 483.
  61. Roman, J. M., G. Sierra, and J. Dukelsky, 2003, Phys. Rev. B 67, 064510.
  62. Rombouts, S., D. Van Neck, and J. Dukelsky, 2003, Phys. Rev. B (to be published).
  63. Schechter, M., Y. Imry, Y. Levinson, and J. von Delft, 2001, Phys. Rev. B 63, 214518.
  64. Shastry, B. S., 1988, Phys. Rev. Lett. 60, 639.
  65. Shastry, B. S., and A. Dhar, 2001, J. Phys. A 34, 6197.
  66. Sierra, G., 2000, Nucl. Phys. B 572, 517.
  67. Sierra, G., 2002, in Statistical Field Theories, edited by A. Cappelli and G. Mussardo, NATO Sciences Series, Vol. 73 (Kluwer Academic, London), pp. 317–328.
  68. Sierra, G., J. Dukelsky, G. G. Dussel, J. von Delft, and F. Braun, 2000, Phys. Rev. B 61, R11 890.
  69. Sigrist, M., and K. Ueda, 1991, Rev. Mod. Phys. 63, 239.
  70. Smith, R. A., and V. Ambegaokar, 1996, Phys. Rev. Lett. 77, 4962.
  71. Sutherland, B., 1971, J. Math. Phys. 12, 246.
  72. Tomonaga, S., 1950, Prog. Theor. Phys. 5, 544.
  73. Volya, A., V. Zelevinsky, and B. A. Brown, 2002, Phys. Rev. C 65, 054312.
  74. von Delft, J., D. S. Golubev, W. Tichy, and A. D. Zaikin, 1996, Phys. Rev. Lett. 77, 3189.
  75. von Delft, J., and R. Poghossian, 2002, Phys. Rev. B 66, 134502.
  76. von Delft, J., and D. C. Ralph, 2001, Phys. Rep. 345, 61.
  77. Wu, L.-A., M. W. Guidry, Y. Sun, and C.-L. Wu, 2002, Phys. Rev. B 67, 014515.
  78. Yuzbashyan, E. A., A. A. Baytin, and B. L. Altshuler, 2003, Phys. Rev. B 68, 214509.
  79. Zhang, S.-C., 1997, Science 275, 1089.
  80. Zhou, H.-Q., J. Links, R. H. McKenzie, and M. D. Gould, 2002, Phys. Rev. B 65, 060502.

Outline

Information

Rev. Mod. Phys. 76, 643– Published 6 August, 2004

    © 2004 American Physical Society

    Sign In to Your Journals Account

    Filter

    Filter

    Article Lookup

    Enter a citation