Skip to main content
SpringerLink
Log in

Valence Bond Ground States in Isotropic Quantum Antiferromagnets

  • Chapter
Condensed Matter Physics and Exactly Soluble Models

Abstract

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.

Supported in part by N.S.F. Grant PHY-80-19754. Fellow of the A.P. Sloan Foundation and the Canadian Institute for Advanced Research

N.S.F. Post-doctoral Fellow

Supported in part by N.S.F. Grant PHY-85-15288-A01

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)

    Article  ADS  Google Scholar 

  2. Affleck, I.: Large-n limit of SU(n) quantum spin chains. Phys. Rev. Lett. 54 966–969 (1985)

    Google Scholar 

  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. Aftleck, L, Haldane, F.D.M.: Critical theory of quantum spin chains. To appear in Phys. Rev. B.

    Google Scholar 

  5. Affleck, I., Lieb, E.H.: A proof of part of Haldane’s conjecture on spin chains. Lett. Math. Phys. 12, 57–69 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  7. Anderson, P.: The s= 1/2 antiferromagnetic ground state: Néel antiferromagnet or quantum liquid. Mat. Res. Bull. 8 153 (1973)

    Google Scholar 

  8. Anderson, P.: The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987)

    Article  ADS  Google Scholar 

  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. Babudjian, J.: Exact solution of the isotropic Heisenberg chain with arbitrary spins: thermodynamics of the model. Nucl. Phys. B 215, 317–336 (1983)

    Article  ADS  Google Scholar 

  11. Bethe, H.: Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. 71, 205–226 (1931)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  13. Bonner, J.C.: Private communication

    Google Scholar 

  14. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. II. Berlin, Heidelberg, New York: Springer 1981

    Google Scholar 

  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. 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)

    Article  ADS  Google Scholar 

  17. Caspers, W.J.: Exact ground states for a class of linear antiferromagnetic spin systems. Physica 115 A, 275–280 (1982)

    Google Scholar 

  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. 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. 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. 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)

    Article  MathSciNet  ADS  Google Scholar 

  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. 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. 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)

    Article  MathSciNet  ADS  Google Scholar 

  25. Haldane, F.D.M.: “O physics” and quantum spin chains (abstract). J. Appl. Phys. 57, 3359 (1985)

    Article  ADS  Google Scholar 

  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. Klein, DJ.: Exact ground states for a class of antiferromagnetic Heisenberg models with short-range interactions. J. Phys. A 15, 661–671 (1982)

    Article  ADS  Google Scholar 

  28. Kulish, P.P., Reshetikhin, N.Yu., Sklyanin, E.K.: Yang-Baxter equation and representation theory. L Lett. Math. Phys. 5, 393–403 (1981)

    MathSciNet  ADS  MATH  Google Scholar 

  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 1982

    Chapter  Google Scholar 

  30. Lieb, E.H., Mattis, D.J.: Ordering energy levels of interacting spin systems. J. Math. Phys. 3, 749–751 (1962)

    Article  ADS  MATH  Google Scholar 

  31. Majumdar, C.K., Ghosh, D.K.: On next nearest-neighbor interaction in linear chain. I, II. J. Math. Phys. 10, 1388–1402 (1969)

    Article  MathSciNet  ADS  Google Scholar 

  32. Majumdar, C.K.: Antiferromagnetic model with known ground state. J. Phys. C 3, 911–915 (1970)

    Article  ADS  Google Scholar 

  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. 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)

    Article  ADS  Google Scholar 

  35. Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. New York: Academic Press 1972

    Google Scholar 

  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. 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. Shastry, B.S., Sutherland, B.: Exact ground state of a quantum mechanical antiferromagnet. Physica 108 B, 1069–1070 (1981)

    Google Scholar 

  39. Shastry, B.S., Sutherland, B.: Excitation spectrum of a dimerized next-neighbor antiferromagnetic chain. Phys. Rev. Lett. 47, 964–967 (1981)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  41. Solyom, J.: Competing bilinear and biquadratic exchange couplings in spin 1 Heisenberg chains. Institute Laue-Langevin preprint

    Google Scholar 

  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. Wells, A.F.: Three-dimensional nets and polyhedra. New York: Wiley 1977

    Google Scholar 

  44. Wreszinsky, W.F.: Charges and symmetries in quantum theories without locality. Fortschr. Phys. 35, 379–413 (1987)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  46. Wreszinski, W.F.: Goldstone’s theorem for quantum spin systems of finite range. J. Math. Phys. 17, 109–111 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  47. Arovas, D.P., Auerbach, A., Haldane, F.D.M.: Extended Heisenberg models of antiferromagnetism: Analogies to the quantum Hall effect. University of Chicago preprint

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of British Columbia, Vancouver, B.C, V6T2A6, Canada

    Ian Affleck

  2. Department of Physics, Princeton University, P.O. Box 708, Princeton, NJ, 08544, USA

    Tom Kennedy, Elliott H. Lieb & Hal Tasaki

Authors
  1. Ian Affleck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tom Kennedy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elliott H. Lieb
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hal Tasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of California, One Shields Ave., 95616-8633, Davis, CA, USA

    Bruno Nachtergaele

  2. Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Jan Philip Solovej

  3. Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, 1090, Wien, Austria

    Jakob Yngvason

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H. (1988). Valence Bond Ground States in Isotropic Quantum Antiferromagnets. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds) Condensed Matter Physics and Exactly Soluble Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06390-3_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-06390-3_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-06093-9

  • Online ISBN: 978-3-662-06390-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation