Valence Bond Ground States in Isotropic Quantum Antiferromagnets

  • Ian Affleck
  • Tom Kennedy
  • Elliott H. Lieb
  • Hal Tasaki


Haldane predicted that the isotropic quantum Heisenberg spin chain is in a “massive” phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exact SO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.


Hexagonal Lattice Cayley Tree Finite Chain Point Correlation Function Infinite Volume 
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  1. 1.
    Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799–802 (1987)ADSCrossRefGoogle Scholar
  2. 2.
    Affleck, I.: Large-n limit of SU(n) quantum spin chains. Phys. Rev. Lett. 54 966–969 (1985)Google Scholar
  3. 3.
    Affleck, I.: Exact critical exponents for quantum spin chains, nonlinear a-models at 6 =7t and the quantum Hall effect. Nucl. Phys. B 265 409–447 (1986)Google Scholar
  4. 4.
    Aftleck, L, Haldane, F.D.M.: Critical theory of quantum spin chains. To appear in Phys. Rev. B.Google Scholar
  5. 5.
    Affleck, I., Lieb, E.H.: A proof of part of Haldane’s conjecture on spin chains. Lett. Math. Phys. 12, 57–69 (1986)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Aizenman, M., Lieb, E.H.: The third law of thermodynamics and the degeneracy of the ground state for lattice systems. J. Stat. Phys. 24, 279–297 (1981)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Anderson, P.: The s= 1/2 antiferromagnetic ground state: Néel antiferromagnet or quantum liquid. Mat. Res. Bull. 8 153 (1973)Google Scholar
  8. 8.
    Anderson, P.: The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987)ADSCrossRefGoogle Scholar
  9. 9.
    Babudjian, J.: Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spins S. Phys. Lett. 90 A, 479–482 (1982)Google Scholar
  10. 10.
    Babudjian, J.: Exact solution of the isotropic Heisenberg chain with arbitrary spins: thermodynamics of the model. Nucl. Phys. B 215, 317–336 (1983)ADSCrossRefGoogle Scholar
  11. 11.
    Bethe, H.: Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. 71, 205–226 (1931)ADSCrossRefGoogle Scholar
  12. 12.
    Blote, H.J., Nightingale, M.P.: Gap of the linear spin-1 Heisenberg antiferromagnet: a Monte-Carlo calculation. Phys. Rev. B 33, 659–661 (1986)ADSCrossRefGoogle Scholar
  13. 13.
    Bonner, J.C.: Private communicationGoogle Scholar
  14. 14.
    Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. II. Berlin, Heidelberg, New York: Springer 1981Google Scholar
  15. 15.
    van den Broek, P.M.: Exact value of the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest and next-nearest neighbour interactions. Phys. Lett. 77 A, 261–262 (1980)Google Scholar
  16. 16.
    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, 371–374 (1986)ADSCrossRefGoogle Scholar
  17. 17.
    Caspers, W.J.: Exact ground states for a class of linear antiferromagnetic spin systems. Physica 115 A, 275–280 (1982)Google Scholar
  18. 18.
    Caspers, WJ., Magnus, W.: Exact ground states for a class of linear quantum spin systems. Physica 119 A, 291–294 (1983)Google Scholar
  19. 19.
    Caspers, W.J., Magnus, W.: Some exact excited states in a linear antiferromagnetic spin system. Phys. Lett. 88 A, 103–105 (1982)Google Scholar
  20. 20.
    Chang, K.: Calculation of the singlet-singlet gap in spin-1, antiferromagnetic quantum spin chains using valence bond diagrams. Senior thesis, Princeton University (1987)Google Scholar
  21. 21.
    Dyson, F.J., Lieb, E.H., Simon, B.: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18, 335–383 (1978)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions, and continuous symmetry breaking. Commun. Math. Phys. 50, 79–95 (1976)Google Scholar
  23. 23.
    Haldane, F.D.M.: Continuum dynamics of the 1—d Heisenberg antiferromagnet: identification with the 0(3) nonlinear sigma model. Phys. Lett. 93 A, 464–468 (1983)Google Scholar
  24. 24.
    Haldane, F.D.M.: Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solutions of the one-dimensional easy-axis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Haldane, F.D.M.: “O physics” and quantum spin chains (abstract). J. Appl. Phys. 57, 3359 (1985)ADSCrossRefGoogle Scholar
  26. 26.
    Jordlo Neves, E., Fernando Perez, J.: Long range order in the ground state of two-dimensional antiferromagnets. Phys. Lett. 114 A, 331–333 (1986)Google Scholar
  27. 27.
    Klein, DJ.: Exact ground states for a class of antiferromagnetic Heisenberg models with short-range interactions. J. Phys. A 15, 661–671 (1982)ADSCrossRefGoogle Scholar
  28. 28.
    Kulish, P.P., Reshetikhin, N.Yu., Sklyanin, E.K.: Yang-Baxter equation and representation theory. L Lett. Math. Phys. 5, 393–403 (1981)MathSciNetADSzbMATHGoogle Scholar
  29. 29.
    Kulish, P.P., Sklyanin, E.K.: Quantum spectral transform method: recent developments. In: Integrable quantum field theories. Ehlers, J., Hepp, K., Kippenhahn, R., Weidenmullen, H.A., Zittartz, J. (eds.). Lecture Notes in Physics, Vol. 151, pp. 61–119. Berlin, Heidelberg, New York: Springer 1982CrossRefGoogle Scholar
  30. 30.
    Lieb, E.H., Mattis, D.J.: Ordering energy levels of interacting spin systems. J. Math. Phys. 3, 749–751 (1962)ADSzbMATHCrossRefGoogle Scholar
  31. 31.
    Majumdar, C.K., Ghosh, D.K.: On next nearest-neighbor interaction in linear chain. I, II. J. Math. Phys. 10, 1388–1402 (1969)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Majumdar, C.K.: Antiferromagnetic model with known ground state. J. Phys. C 3, 911–915 (1970)ADSCrossRefGoogle Scholar
  33. 33.
    Nienhuis, B.: Exact critical point and critical exponents of 0(n) models in two dimensions. Phys. Rev. Lett. 49 1062–1064 (1982)Google Scholar
  34. 34.
    Oitmaa, J., Parkinson, J.B., Bonner, J.C.: Crossover effects in a general spin-1-bilinearbiquadratic exchange Hamiltonian. J. Phys. C 19, L595 — L601 (1986)ADSCrossRefGoogle Scholar
  35. 35.
    Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. New York: Academic Press 1972Google Scholar
  36. 36.
    Renard, J.P., Verdaguer, M., Regnault, L.P., Erkelens, W.C., Rossatmignod, J., Stirling, W.G.: Presumption for a quantum energy gap in the quasi-one-dimensional s=1 Heisenberg antiferromagnet Ni[C2H$N2]ZNO2[C1O4]. To appear in Europhys. Lett.Google Scholar
  37. 37.
    Schulz, Hi., Ziman, T.: Finite-length calculations of ry and phase diagrams of quantum spin chains. Phys. Rev. B 33, 6545–6548 (1986)Google Scholar
  38. 38.
    Shastry, B.S., Sutherland, B.: Exact ground state of a quantum mechanical antiferromagnet. Physica 108 B, 1069–1070 (1981)Google Scholar
  39. 39.
    Shastry, B.S., Sutherland, B.: Excitation spectrum of a dimerized next-neighbor antiferromagnetic chain. Phys. Rev. Lett. 47, 964–967 (1981)ADSCrossRefGoogle Scholar
  40. 40.
    Shastry, B.S., Sutherland, B.: Exact solution of a large class of interacting quantum systems exhibiting ground state singularities. J. Stat. Phys. 33, 477–484 (1983)MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    Solyom, J.: Competing bilinear and biquadratic exchange couplings in spin 1 Heisenberg chains. Institute Laue-Langevin preprintGoogle Scholar
  42. 42.
    Takhtajan, L.: The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins. Phys. Lett. 87 A, 479–482 (1982)Google Scholar
  43. 43.
    Wells, A.F.: Three-dimensional nets and polyhedra. New York: Wiley 1977Google Scholar
  44. 44.
    Wreszinsky, W.F.: Charges and symmetries in quantum theories without locality. Fortschr. Phys. 35, 379–413 (1987)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Landau, L., Fernando-Perez, J., Wreszinski, W.F.: Energy gap, clustering, and the Goldstone theorem in statistical mechanics. J. Stat. Phys. 26, 755–766 (1981)ADSCrossRefGoogle Scholar
  46. 46.
    Wreszinski, W.F.: Goldstone’s theorem for quantum spin systems of finite range. J. Math. Phys. 17, 109–111 (1976)MathSciNetADSCrossRefGoogle Scholar
  47. 47.
    Arovas, D.P., Auerbach, A., Haldane, F.D.M.: Extended Heisenberg models of antiferromagnetism: Analogies to the quantum Hall effect. University of Chicago preprintGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Ian Affleck
    • 1
  • Tom Kennedy
    • 2
  • Elliott H. Lieb
    • 2
  • Hal Tasaki
    • 2
  1. 1.Department of PhysicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of PhysicsPrinceton UniversityPrincetonUSA

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