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

Physical Review Journals Archive

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Phase Transition in the Two-Dimensional Heisenberg Ferromagnet

Vinod Mubayi*,†

Robert V. Lange‡

  • Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14850

  • Brandeis University, Waltham, Massachusetts 02154

  • *Work supported in part by the Advanced Research Projects Agency through the Materials Science Center, Cornell University, Report No. 958.
  • †This work is based, in part, on the thesis submitted by Vinod Mubayi for the Ph.D. degree in Physics to Brandeis University in 1967.
  • ‡Work supported in part by the National Science Foundation under contract Nos. GP-5374 and GP-7397.
PDF

Phys. Rev. 178, 882 – Published 10 February, 1969

DOI: https://doi.org/10.1103/PhysRev.178.882

Abstract

We develop a Green-function theory to describe the thermodynamic behavior of a plane square lattice with spins of magnitude one-half located at the lattice sites interacting via a nearest-neighbor Heisenberg ferromagnetic coupling. Our approximation technique involves a decoupling of the hierarchy of Green-function equations similar in some respects to that found in the random-phase approximation (RPA) but improved to include spin correlations neglected in the RPA. Such an improvement is essential for the two-dimensional problem. Our theory predicts a phase transition at the temperature given by kTc=2J, where J is the exchange parameter. As T approaches Tc from above, the static susceptibility diverges as 1(T−Tc). The spontaneous magnetization is zero at all nonzero temperatures, both above and below the critical point. Therefore, our theory is consistent with the existing rigorous proof of Mermin and Wagner that the spontaneous magnetization must be zero for T≠0, and displays the divergent susceptibility predicted by Stanley and Kaplan from an analysis of high-temperature expansions for related two-dimensional spin systems.

View figure in article

Comments & Replies

Comment on "Phase Transition in the Two-Dimensional Ferromagnet"

Richard P. Kenan
Phys. Rev. B 1, 3205 (1970)

References (22)

  1. H. E. Stanley and T. A. Kaplan, Phys. Rev. Letters 17, 913 (1966) J. Appl. Phys. 48, 975 (1967)
  2. G. S. Rushbrooke and P. J. Wood, Proc. Phys. Soc. (London) A68, 1161 (1955) Mol. Phys. 1, 257 (1958) H. E. Stanley and T. A. Kaplan, J. Appl. Phys. 38, 977 (1967)
  3. N. D. Mermin and H. Wagner, Phys. Rev. Letters 17, 1133 (1966)
  4. P. C. Hohenberg, Phys. Rev. 258, 383 (1967)
  5. N. N. Bogoliubov, Physik. Abh. Sowjetunion 6, 1 (1962) ibid.6, 113 (1962) ibid.6, 229 (1962)
  6. H. Wagner, Z. Physik 195, 273 (1966) R. V. Lange, Phys. Rev. 146, 310 (1966)
  7. N. N. Bogoliubov and S. V. Tyablikov, Dokl. Akad. Nauk SSSR 126, 53 (1959) [English transl.: Soviet Phys.—Doklady 4, 589 (1959)] F. Englert, Phys. Rev. Letters 5, 103 (1960) R. Brout [Phase Transitions (W. A. Benjamin, Inc., New York, 1965)]
  8. Sec. II A R. B. Griffiths, Phys. Rev. 152, 240 (1966)
  9. J. Koppen, R. Hamersma, J. V. Lebesque, and A. R. Miedema, Phys. Letters 25A, 376 (1967)
  10. D. N. Zubarev, Usp. Fiz. Nauk 71, 71 (1960) [English transl.: Soviet Phys.—Usp. 3, 320 (1960)] V. L. Bonch-Bruevich and S. V. Tyablikov, in The Green Function Method in Statistical Mechanics, edited by D. ter Haar (North-Holland Publishing Company, Amsterdam, 1962)
  11. H. B. Callen, Phys. Rev. 230, 590 (1963)

  12. Omitted endnote

  13. E. Jahnke and F. Emde, Tables of Functions (Dover Publications, Inc., New York, 1943)
  14. [13]
  15. H. B. Dwight, Tables of Integrals and Other Mathematical Data (The Macmillan Company, New York, 1963)
  16. Handbook of Mathematical Functions, edited by Milton Abramovitz and I. A. Stegun (Dover Publications, Inc., New York, 1965)
  17. G. N. Watson, Theory of Bessel Functions (Cambridge University Press, New York, 1958)
  18. M. D. Girardeau, J. Math. Phys. 6, 1083 (1965)
  19. S. P. Heims, Phys. Rev. Letters 14, 850 (1965)
  20. Dr. V. Korenman (private communication)
  21. S. V. Tyablikov, Methods in the Quantum Theory of Magnetism (Plenum Press, Inc., New York, 1967)
  22. F. J. Dyson, lecture given at the Brandeis Summer Institute for Theoretical Physics, 1966 (unpublished)

Outline

Information

Phys. Rev. 178, 882– Published 10 February, 1969
  • Received 11 June 1968

© 1969 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation