Journal of Soviet Mathematics

, Volume 24, Issue 2, pp 241–267 | Cite as

Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model

  • L. D. Faddeev
  • L. A. Takhtadzhyan

Abstract

The work gives a consistent and uniform exposition of all known results related to Heisenberg model. The classification of excitations is presented and their scattering is described both in ferromagnetic and the antiferromagnetic cases. It is shown that in the antiferromagnetic case there exists only one excitation with spin 1/2 which is a kink in the following sense: in physical states there is only an even number of kinks-spin waves, therefore they always have an integer spin. Thus, it is shown that the conventional picture of excitations Is wrong in the antiferromagnetic case and the spin wave has spin 1/2, matrix is calculated.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature cited

  1. 1.
    W. Heisenberg, “Zur Theorie des Ferromagnetismus,” Z. Phys.,49, 619–636 (1928).Google Scholar
  2. 2.
    H. Bethe, “Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette,” Z. Phys.,71, 205–226 (1931).Google Scholar
  3. 3.
    L. Hulthen, “Über das Austauschproblem eines Kristalles,” Arkiv Mat., Astron. Fysik,26A, No. 11, 1–106 (1938).Google Scholar
  4. 4.
    J. Des Cloizeaux and J. J. Pearson, “Spin-wave spectrum of the antiferromagnetic linear chain,” Phys. Rev.,128, No. 5, 2131–2135 (1962).Google Scholar
  5. 5.
    R. Orbach, “Linear antiferromagnetic chain with anisotropic coupling,” Phys. Rev.,112, No. 2, 309–316 (1958).Google Scholar
  6. 6.
    C. N. Yang and C. P. Yang, “One-dimensional chain of anisotropic spin-spin interaction. I. Proof of Bethe's hypothesis for ground state in finite system,” Phys. Rev.,150, No. 1, 321–327 (1966).Google Scholar
  7. 7.
    C. N. Yang and C. P. Yang, “One-dimensional chain of anisotropic spin-spin interaction. II. Properties of the ground state energy per lattice site for an finite system,” Phys. Rev.,150, No. 1, 327–339 (1966).Google Scholar
  8. 8.
    R. J. Baxter, “One-dimensional anisotropic Heisenberg chain,” Ann. Phys.,70, No. 2, 323–327 (1972).Google Scholar
  9. 9.
    M. Gaudin, “Modeles exacts en mecanique statistique: La methode de Bethe et ses generalisations,” Centre d'Etudes Nucleaires de Saclay, CEA-N-1559(1), Septembre, 1972.Google Scholar
  10. 10.
    M. Takahashi, “One-dimensional Heisenberg model at finite temperatures,” Progr. Theor. Phys., 46, No. 2, 401–415 (1971).Google Scholar
  11. 11.
    A. A. Ovchinnikov, “The spectrum of excitations of the antiferromagnetic Heisenberg chain,” Zh. Eksp. Teor. Fiz.,56, No. 4, 1354–1365 (1969).Google Scholar
  12. 12.
    P. P. Kulish and N. Yu. Reshetikhin, “Generalized Heisenberg ferromagnet and Gross-Neveu model,” Zh. Eksp. Teor. Fiz.,80, No. 1, 214–228 (1981).Google Scholar
  13. 13.
    L. D. Faddeev, “Quantum completely integrable models in field theory,” Soviet Scientific Reviews, Harvard Academic, London, 107–155 (1980).Google Scholar
  14. 14.
    E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, “Quantum method of the inverse problem,” Teor. Mat. Fiz.,40, No. 2, 194–220 (1979).Google Scholar
  15. 15.
    L. A. Takhtadzhyan and L. D. Faddeev, “The quantum method of the inverse problem and the Heisenberg XYZ model,” Usp. Mat. Nauk,34, No. 5, 13–63 (1979).Google Scholar
  16. 16.
    A. A. Belavin, “Exact solution of the two-dimensional model with asymptotic freedom,” Phys. Lett.,87B, Nos. 1–2, 117–121 (1979).Google Scholar
  17. 17.
    N. Andrei and J. H. Lowenstein, “Diagonalization of the chiral invariant Gross-Neveu Hamiltonian,” Phys. Rev. Lett.,43, No. 23, 1698–1700 (1979).Google Scholar
  18. 18.
    N. Andrei and J. H. Lowenstein, “Derivation of the chiral Gross-Neveu spectrum for arbitrary SU(N)-symmetry,” Phys. Lett.,90B, No. 3, 106–110 (1980).Google Scholar
  19. 19.
    M. Gaudin, “Etude d'un modele a une dimension pour un system de fermions en interaction,” Centre d'Etude Nucleaires de Saclay, Rapport CEA-R-3569 (1968).Google Scholar
  20. 20.
    C. N. Yang, “Some exact results for the many-body problem in one dimension with repulsive delta-function interaction,” Phys. Rev. Lett.,19, No. 23, 1312–1314 (1967).Google Scholar
  21. 21.
    E. H. Lieb and V. F. Wu, “Absence of Mott transition in an exact solution of shortrange 1-band model in 1-dimension,” Phys. Rev. Lett.,20, No. 25, 1445–1448 (1968).Google Scholar
  22. 22.
    J. Hubbard, “Electron correlations in narrow energy bands,” Proc. R. Soc.,A276, No. 1365, 238–257 (1963).Google Scholar
  23. 23.
    P. P. Kulish and E. K. Sklyanin, “Quantum inverse scattering method and the Heisenberg ferromagnet,” Phys. Lett. A,70A, Nos. 5–6, 461–463 (1979).Google Scholar
  24. 24.
    M. Gaudin, B. M. McCoy, and T. T. Wu, “Normalization sum for the Bethe's hypothesis wave functions of the Heisenberg-Ising chain,” Phys. Rev. D,23, No. 2, 417–419 (1981).Google Scholar
  25. 25.
    J. von Neumann, “On infinite direct products,” Comp. Math.,6, No. 1, 1–77 (1938).Google Scholar
  26. 26.
    E. K. Sklyanin, “Quantum version of the method of the inverse scattering problem,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,95, 55–128 (1980).Google Scholar
  27. 27.
    E. K. Sklyanin and L. D. Faddeev, “Quantum-mechanical approach to completely integrable models of field theory,” Dokl. Akad. Nauk SSSR,243, 1430–1433 (1978).Google Scholar
  28. 28.
    L. E. Thomas, “Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. I,” J. Math. Anal. Appl.,59, No. 2, 392–414 (1977).Google Scholar
  29. 29.
    D. Babbitt and L. Thomas, “Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. II. An explicit Plancherel formula,” Commun. Math. Phys.,54, No. 3, 255–278 (1977).Google Scholar
  30. 30.
    D. Babbitt and L. Thomas, “Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. III. Scattering theory,” J. Math. Phys.,19, No. 8, 1699–1704 (1978).Google Scholar
  31. 31.
    A. B. Zamolodchikov and A. B. Zamolodchikov, “Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models,” Ann. Phys.,120, No. 2, 253–291 (1979).Google Scholar
  32. 32.
    V. E. Korepin, “Direct calculation of the S-matrix in the thirring model,” Teor. Mat. Fiz.,40, No. 2, 169–189 (1979).Google Scholar
  33. 33.
    P. Fazekas and A. Süto, “On the spectrum of singlet excitations of the S=1/2 linear Heisenberg antiferromagnet,” Solid-State Commun.,19, No. 11, 1045–1048 (1976).Google Scholar
  34. 34.
    A. Süto, “Excitations in the antiferromagnetic Heisenberg chain,” Solid-State Commun.,20, No. 11, 681–682 (1976).Google Scholar
  35. 35.
    H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York (1955).Google Scholar
  36. 36.
    B. Berg, M. Karowskii, P. Weisz, and V. Kurak, “Factorized U(n) symmetric S-matrices in two dimensions,” Nucl. Phys.,B134, No. 1, 125–132 (1978).Google Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • L. D. Faddeev
  • L. A. Takhtadzhyan

There are no affiliations available

Personalised recommendations