Skip to main content

Order and disorder lines in systems with competing interactions: I. Quantum spins atT=0

Abstract

In the parameter space of systems with competing interactions there are specific trajectories called order (disorder) lines. Along these trajectories the competition between the different interactions effectively reduces the dimensionality of the system and the model can be exactly solved. It is shown that the order (disorder) trajectories end up at a multicritical point. The method of Peschel and Emery is used to determine the (anisotropic) critical behavior of the spin-spin correlation functions near the multicritical point. The quantum spin systems discussed here include theXYZ chain in a field, the straggeredXYZ chain in a field, and a Hamiltonian version of a three-dimensional Ising model with biaxial competing interactions.

This is a preview of subscription content, access via your institution.

References

  1. I. Peschel and V. J. Emery,Z. Phys. B 43:241 (1981). For a different approach see also J. Kurman, H. Thomas, and G. Müller,Physica A, to be published.

    Google Scholar 

  2. J. Stevenson,Can. J. Phys. 48:2118 (1970);J. Math. Phys. 11:420 (1970).

    Google Scholar 

  3. M. B. Geilikmann,Sov. Phys. JETP 39:570 (1974); E. Fradkin and T. P. Eggarter,Phys. Rev. A 14:495 (1976).

    Google Scholar 

  4. E. Barouch and B. McCoy,Phys. Rev. A 3:786 (1971).

    Google Scholar 

  5. M. Steiner, J. Villain, and G. G. Windsor,Adv. Phys. 25:87 (1976).

    Google Scholar 

  6. J. C. Bonner, H. W. J. Blöte, H. Beck, and G. Müller, inPhysics in One Dimension, J. Bernasconi and T. Schneider, eds., Springer-Verlag, New York (1981).

    Google Scholar 

  7. J. T. Devreese, R. P. Evrard, and V. E. van Doren, eds.,Highly Conducting One-Dimensional Solids, Plenum Press, New York (1980).

    Google Scholar 

  8. B. Sutherland,J. Math. Phys. 11:3183 (1970).

    Google Scholar 

  9. R. J. Baxter,Ann. Phys. (N.Y.) 70:193, 323 (1972).

    Google Scholar 

  10. E. Fradkin and L. Susskind,Phys. Rev. D 17:2637 (1978).

    Google Scholar 

  11. For a review see P. W. Kasteleyn, inFundamental Problems in Statistical Mechanics, Vol. 3, E. D. G. Cohen, ed., North-Holland, Amsterdam (1975).

    Google Scholar 

  12. For a recent review see P. C. Hohenberg and B. I. Halperin,Rev. Mod. Phys. 49:435 (1977).

    Google Scholar 

  13. R. Glauber,J. Math. Phys. 4:294 (1963).

    Google Scholar 

  14. See K. Kawasaki, inPhase Transitions and Critical Phenomena, Vol. II, C. Domb and M. S. Green, eds., Academic, New York (1972).

    Google Scholar 

  15. B. U. Felderhof,Rep. Math. Phys. 1:215 (1970); E. D. Siggia,Phys. Rev. B 16:2319 (1977).

    Google Scholar 

  16. H. A. Kramers and G. H. Wannier,Phys. Rev. 60:252 (1941); L. Mittag and M. J. Stephen,J. Math. Phys. 12:441 (1971).

    Google Scholar 

  17. R. J. Baxter,Ann. Phys. (N. Y.) 76(1), 25, 48 (1973).

    Google Scholar 

  18. J. D. Johnson, S. Krinski, and B. M. McCoy,Phys. Rev. A 8:2526 (1973).

    Google Scholar 

  19. B. M. McCoy and T. T. Wu,Nuovo Cimento 56B:311 (1968).

    Google Scholar 

  20. C. N. Yang and C. P. Yang,Phys. Rev. 150:321 (1966).

    Google Scholar 

  21. J. D. Johnson and B. M. McCoy,Phys. Rev. A 4:1613 (1972).

    Google Scholar 

  22. E. Lieb, T. Schultz, and D. Mattis,Ann. Phys. (N. Y.) 16:407 (1961).

    Google Scholar 

  23. S. Katsura,Phys. Rev. 127:1508 (1962).

    Google Scholar 

  24. T. Niemeijer,Physica 36:377 (1967).

    Google Scholar 

  25. M. Suzuki,Progr. Theor. Phys. 46:1377 (1971).

    Google Scholar 

  26. M. J. Stephen and L. Mittag,J. Math. Phys. 13:1944 (1972).

    Google Scholar 

  27. W. Selke and M. E. Fisher,Z. Phys. B 40:71 (1980).

    Google Scholar 

  28. S. Redner,J. Stat. Phys. 25:15 (1981).

    Google Scholar 

  29. F. Haake and K. Thol,Z. Phys. B 40:219 (1980); R. Pandit, G. Forgács, and P. Ruján,Phys. Rev. B 24:1576 (1981).

    Google Scholar 

  30. R. M. Hornreich, R. Liebman, H. G. Schuster, and W. Selke,Z. Phys. B 35:91 (1979).

    Google Scholar 

  31. For a review of ferroelectric models see E. H. Lieb and F. Y. Wu, inPhase Transitions and Critical Phenomena, Vol. II, C. Domb and M. S. Green, eds., Academic Press, London (1972); J. L. Black and V. J. Emery,Phys. Rev. B 23:429 (1981); M. Kohmoto, M. P. M. den Nijs, and L. P. Kandanoff,Phys. Rev. B 24:5229 (1981); C. J. Hamer,J. Phys. A 14:2981 (1981).

    Google Scholar 

  32. Z. Rácz and M. F. Collins,Phys. Rev. B 13:3074 (1973).

    Google Scholar 

  33. P. Ruján,Phys. Rev. B 24:6620 (1981); M. N. Barber and P. M. Duxbury,J. Phys. A 14:L251 (1981).

    Google Scholar 

  34. P. Ruján, in preparation.

Download references

Author information

Authors and Affiliations

Authors

Additional information

On leave from and address after September 1, 1982: Institute for Theoretical Physics, Eötvös University, Puskin U. 5-7, 1088 Budapest, Hungary.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ruján, P. Order and disorder lines in systems with competing interactions: I. Quantum spins atT=0. J Stat Phys 29, 231–245 (1982). https://doi.org/10.1007/BF01020784

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01020784

Key words

  • Quantum spin systems
  • order
  • disorder
  • kinetic Ising model
  • duality transformation
  • Hamiltonian limit