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Hidden symmetry breaking and the Haldane phase inS=1 quantum spin chains

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Abstract

We study the phase diagram ofS=1 antiferromagnetic chains with particular emphasis on the Haldane phase. The hidden symmetry breaking measured by the string order parameter of den Nijs and Rommelse can be transformed into an explicit breaking of aZ 2×Z 2 symmetry by a nonlocal unitary transformation of the chain. For a particular class of Hamiltonians which includes the usual Heisenberg Hamiltonian, we prove that the usual Néel order parameter is always less than or equal to the string order parameter. We give a general treatment of rigorous perturbation theory for the ground state of quantum spin systems which are small perturbations of diagonal Hamiltonians. We then extend this rigorous perturbation theory to a class of “diagonally dominant” Hamiltonians. Using this theory we prove the existence of the Haldane phase in an open subset of the parameter space of a particular class of Hamiltonians by showing that the string order parameter does not vanish and the hiddenZ 2×Z 2 symmetry is completely broken. While this open subset does not include the usual Heisenberg Hamiltonian, it does include models other than VBS models.

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References

  1. Albanes, C.: On the spectrum of the Heisenberg Hamiltonian. J. Stat. Phys.55, 297 (1989)

    Google Scholar 

  2. Affleck, I.: Quantum spin chains and the Haldane gap. J. Phys.: Condensed Matter1, 3047 (1989)

    Google Scholar 

  3. Affleck, I., Halperin, B.I.: private communication

  4. Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett.59, 799 (1987); Valence-bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys.115, 477 (1988)

    Google Scholar 

  5. Affleck, I., Lieb, E.H.: A proof of part of Haldane's conjecture on spin chains. Lett. Math. Phys.12, 57 (1986)

    Google Scholar 

  6. Ajiro, Y., Goto, T., Kikuchi, H., Sakakibara, T., Inami, T.: High-field magnetization of a quasi-one-dimensionalS=1 antiferromagnet Ni(C2H8N2)2NO2(ClO4): Observation of the Haldane gap. Phys. Rev. Lett.63, 1424 (1989)

    Google Scholar 

  7. Arovas, D.P., Auerbach, A., Haldane, F.D.M.: Extended Heisenberg models of antiferroagnetism: Analogies to the fractional quantum Hall effect. Phys. Rev. Lett.60, 531 (1988)

    Google Scholar 

  8. Asano, T.: Theorems on the partition functions of the Heisenberg ferromagnets. J. Phys. Soc. Jpn.29, 350 (1970)

    Google Scholar 

  9. Babudjian, H.M.: Exact solutions of the one-dimensional isotropic Heisenberg chain with arbitraryS. Phys. Lett.90A, 479 (1982); Exact solutions of the isotropic Heisenberg chain with arbitrary spins: Thermodynamics of the model. Nucl. Phys.B215, 317 (1983)

    Google Scholar 

  10. Barber, M.N., Batchelor, M.T.: Spectrum of the biquadratic spin-1 antiferromagnetic chain. Phys. Rev.B40, 4621 (1989)

    Google Scholar 

  11. Borgs, C.: Confinement, deconfinement and freezing in lattice Yang Mills theories with continuous time. Commun. Math. Phys.116, 309 (1988)

    Google Scholar 

  12. Botet, R., Julien, R., Kolb, M.: Finite-size scaling study of the spin-1 Heisenberg-Ising chain with uniaxial anisotropy. Phys. Rev.B28, 3914 (1983)

    Google Scholar 

  13. Bratteli, O., Kishimoto, A., Robinson, D.: Ground states of quantum spin systems. Commun. Math. Phys.64, 41 (1978)

    Google Scholar 

  14. Brydges, D.: A short course on cluster expansions. In: Critical Phenomena, Random Systems, Gauge Theories. Osterwalder, K., Stora, R. (eds.), Amsterdam: Elsevier 1986

    Google Scholar 

  15. Buyers, W., Morra, R., Armstrong, R., Hogan, M., Gerlack, P., Hirakawa, K.: Experimental evidence for the Haldane gap in a spin-1, nearly isotropic, antiferromagnetic chain. Phys. Rev. Lett.56, 371 (1986)

    Google Scholar 

  16. den Nijs, M., Rommelse, K.: Preroughning transitions in crystal surfaces and valence-bond phases in quantum spin chains. Phys. Rev.B40, 4709 (1989)

    Google Scholar 

  17. Fannes, M., Nachtergaele, B., Werner, R.F.: Exact antiferromagnetic ground states of quantum spin chains. Europhys. Lett.10, 633 (1989); Finitely correlated states on quantum spin chains. Commun. Math. Phys.144, 443 (1992)

    Google Scholar 

  18. Ginibre, J.: Existence of phase transitions for quantum lattice system. Commun. Math. Phys.14, 205 (1969)

    Google Scholar 

  19. Girvin, S.M., Arovas, D.: Hidden topological order in integer quantum spin chains. Physica ScriptaT27, 156 (1989)

    Google Scholar 

  20. Glarum, S.H., Geschwind, S., Lee, K.M., Kaplan, M.L., Michel, J.: Observation of fractional spinS=1/2 on open ends ofS=1 linear antiferromagnetic chains: Nonmagnetic doping. Phys. Rev. Lett.67, 1614 (1991)

    Google Scholar 

  21. Hagiwara, M., Katsumata, K., Affleck, I., Halperin, B.I., Renard, J.P.: Observation ofS=1/2 degrees of freedom in anS=1 linear-chain Heisenberg antiferromagnet. Phys. Rev. Lett.65, 3181 (1990)

    Google Scholar 

  22. Haldane, F.D.M.: Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with theO(3) nonlinear sigma model. Phys. Lett.93A, 464 (1983); Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the onedimensional easy-axis Néel state. Phys. Rev. Lett.50, 1153 (1983)

    Google Scholar 

  23. Hatsugai, Y., Kohmoto, M.: Numerical study of hidden antiferromagnetic order in the Haldane phase. Phys. Rev. B44, 11789 (1991)

    Google Scholar 

  24. Katsumata, K., Hori, H., Takeuchi, T., Date, M., Yamagishi, A., Renard, J.P.: Magnetization process of anS=1 linear-chain Heisenberg antiferromagnet. Phys. Rev. Lett.63, 86 (1989)

    Google Scholar 

  25. Kennedy, T.: Exact diagonalization of open spin 1 chains. J. Phys.: Condens. Matter2, 5737–5745 (1990)

    Google Scholar 

  26. Kennedy, T., Tasaki, H.: HiddenZ 2×Z 2 symmetry breaking in Haldane gap antiferromagnets. Phys. Rev. B (to appear)

  27. Kirkwood, J.R., Thomas, L.E.: Expansions and phase transitions for the ground state of quantum Ising lattice systems. Commun. Math. Phys.88, 569 (1983)

    Google Scholar 

  28. Klümper, A.: The spectra ofq state vertex models and related antiferromagnetic quantum spin chains. J. Phys. A.: Math. Gen.23, 809 (1990)

    Google Scholar 

  29. Knabe, S.: Energy gap and elementary excitations for certain VBS quantum antiferromagnets. J. Stat. Phys.52, 627 (1988)

    Google Scholar 

  30. Kolb, M.: Symmetry and boundary condition of planar spin systems. Phys. Rev.B31, 7494 (1985)

    Google Scholar 

  31. Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys.103, 491 (1986)

    Google Scholar 

  32. Kulish, P., Reshtikhin, N. Yu, Sklyanin, E.: Yang-Baxter equation and representation theory. Lett. Math. Phys.5, 393 (1981); Quantum spectral transform method: Recent developments. In: Integrable quantum field theories. Ehlers, J., Hepp, K., Kippenhahn, R., Weidenmüllen, H.A., Zittartz, J. (eds.). Lecture Notes in Physics, Vol. 151, pp. 61. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  33. Lai, J.K.: Lattice gas with nearest neighbor interaction in one dimension with arbitrary statistics. J. Math. Phys.15, 1675 (1974)

    Google Scholar 

  34. Liang, S.: Monte Carlo calculation of the correlation functions for Heisenberg Spin Chains at T=0. Phys. Rev. Lett.64, 1597 (1990)

    Google Scholar 

  35. Lieb, E.H., Mattis, D.: Ordering energy levels in interacting spin chains. J. Math. Phys.3, 749–751 (1962)

    Google Scholar 

  36. Lieb, E., Schultz, T., Mattis, D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. (NY)16, 407 (1961)

    Google Scholar 

  37. Matsui, T.: Uniqueness of the translationally invariant ground state in quantum spin systems. Commun. Math. Phys.126, 453 (1990)

    Google Scholar 

  38. Matsui, T.: Remarks on the duality of 1 dimensional quantum spin models (preprint)

  39. Nightingale, M.P., Blöte, H.W.J.: Gap of the linear spin-1 Heisenberg antiferromagnet: A Monte Carlo calculation. Phys. Rev.B33, 659 (1986)

    Google Scholar 

  40. Oitmaa, J., Parkinson, J.B., Bonner, J.C.: Crossover effects in a general spin-1-bilinearbiquadratic exchange Hamiltonian. J. Phys. C19, L595 (1986)

    Google Scholar 

  41. Oshikwa, M.: HiddenZ 2×Z 2 symmetry in arbitrary integerS quantum spin chains. J. Phys.: Condens. Matter (to appear)

  42. Perk, J.H.H., Schultz, C.L.: Diagonalization of the transfer matrix of a nonintersecting string model. Physica122A, 50 (1983)

    Google Scholar 

  43. Renard, J.P., Verdaguer, M., Regnault, L.P., Erkelens, W.A.C., Rossat-Mignod, J., Ribas, J., Stirling, W.G., Vettier, C.: Quantum energy gap in two quasi-one-dimensionalS=1 Heisenberg antiferromagnets. J. Appl. Phys.63, 3538 (1988)

    Google Scholar 

  44. Robinson, D.: A new proof of the existence of phase transitions in the anisotropic Heisenberg model. Commun. Math. Phys.14, 195 (1969)

    Google Scholar 

  45. Sakai, T., Takahashi, M.: Energy gap ofS=1 antiferromagnetic Heisenberg chain. Phys. Rev.42, 1090 (1990)

    Google Scholar 

  46. Schorr, A.J.: Antiferromagnetic quantum spin chains with an anisotropic coupling. Princeton University senior thesis, 1987 (unpublished)

  47. Schulz, H.J.: Phase diagrams and correlation exponents for quantum spin chains of arbitrary spin quantum number. Phys. Rev.B34, 6372 (1986)

    Google Scholar 

  48. Schulz, H.J., Ziman, T.: Finite-length calculation of η and phase diagrams of quantum spin chains. Phys. Rev.B33, 6545 (1986)

    Google Scholar 

  49. Singh, R.P.R., Gelfand, M.P.: Order and criticality in spin-1 chains. Phys. Rev. Lett.61, 2133 (1988)

    Google Scholar 

  50. Sólyom, J.: Competing bilinear and biquadratic exchange couplings in spin 1 Heisenberg chains. Phys. Rev.B36, 8642 (1987)

    Google Scholar 

  51. Steiner, M., Kakurai, K., Kjems, J.K., Petitgrand, D., Pynn, R.: Inelastic neutron scattering studies on 1D near-Heisenberg antiferromagnets: A test of the Haldane conjecture. J. Appl. Phys.61, 3953 (1987)

    Google Scholar 

  52. Sutherland, B.: Model for a multi component quantum system. Phys. Rev.B12, 3795 (1975)

    Google Scholar 

  53. Suzuki, M., Fisher, M.E.: Zeros of the partition function for the Heisenberg, ferroelectric, and general Ising models. J. Math. Phys.12, 235 (1971)

    Google Scholar 

  54. Takada, S., Kubo, K.: Nonlocal unitary transformation onS=1 antiferromagnetic spin chains. J. Phys. Soc. Jpn.60, 4026 (1991)

    Google Scholar 

  55. Takahashi, M.: Monte Carlo calculation of elementary excitation of spin chains. Phys. Rev. Lett.62, 2313 (1989)

    Google Scholar 

  56. Takhatajan, L.: The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins. Phys. Lett.87A, 479 (1982)

    Google Scholar 

  57. Tasaki, H.: Haldane gap in three dimensions: A rigorous example. Phys. Rev. Lett.64, 2066 (1990)

    Google Scholar 

  58. Tasaki, H.: Quantum liquid in antiferromagnetic chains: A stochastic geometric approach to the Haldane gap. Phys. Rev. Lett.66, 798 (1991)

    Google Scholar 

  59. Tasaki, H.: Haldane gap antiferromagnets in transverse magnetic field. J. Phys.: Condens. Matter3, 5875 (1991)

    Google Scholar 

  60. Tasaki, H.: Haldane gap in quantum spin chains. A geometric approach. (in preparation)

  61. Thomas, L.E., Yin, Z.: Low temperature expansions for the Gibbs states of weakly interacting quantum Ising lattice systems. Commun. Math. Phys.91, 405 (1983); Low temperature expansions for the Gibbs states of quantum Ising lattice systems. J. Math. Phys.25, 3128 (1984)

    Google Scholar 

  62. Uimin, G.V.: One-dimensional problem forS=1 with modified antiferromagnetic Hamiltonian. J.E.T.P. Lett.12, 225 (1970)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, 85721, Tucson, AZ, USA

    Tom Kennedy

  2. Department of Physics, Gakushuin University, Mejiro, Toshima-ku, 171, Tokyo, Japan

    Hal Tasaki

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  1. Tom Kennedy
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Communicated by M. Aizenman

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Kennedy, T., Tasaki, H. Hidden symmetry breaking and the Haldane phase inS=1 quantum spin chains. Commun.Math. Phys. 147, 431–484 (1992). https://doi.org/10.1007/BF02097239

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  • DOI: https://doi.org/10.1007/BF02097239

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