Random Walk Analysis of the Commensurate-Incommensurate Transition in the Isotropic Spin-1 Chain

Abstract

It has been observed that in the isotropic spin-1 chain a transition in the asymptotic properties of the correlation function (commensurate-incommensurate transition) occurs at the AKLT point. We propose a simple random-walk-type argument, explaining this transition. Also, we consider a modification of the AKLT model, for which this argument can be turned into a rigorous proof.

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

References

  1. 1.

    Affleck, I., Kennedy, T., Lieb, E.H., Tasaki, H.: Valence bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys. 115, 477–528 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  2. 2.

    Fáth, G., Sütő, A.: Commensurate and incommensurate correlations in Haldane-gap antiferromagnets. Phys. Rev. B 62, 3778–3785 (2000)

    Article  ADS  Google Scholar 

  3. 3.

    Haldane, F.D.M.: Continuum dynamics of the 1-d Heisenberg antiferromagnet: identification with the O(3) nonlinear sigma model. Phys. Lett. A 93, 464–468 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  4. 4.

    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  ADS  MathSciNet  Google Scholar 

  5. 5.

    Kennedy, T.: Ornstein–Zernike decay in the ground state of the quantum Ising model in a transverse magnetic field. Commun. Math. Phys. 137, 599–615 (1991)

    MATH  Article  ADS  Google Scholar 

  6. 6.

    Kennedy, T., Tasaki, H.: Hidden symmetry breaking and the Haldane phase in S=1 quantum spin chains. Commun. Math. Phys. 147, 431–484 (1992)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  7. 7.

    Murashima, T., Nomura, K.: Incommensurability and edge states in the one-dimensional S=1 bilinear-biquadratic model. Phys. Rev. B 73, 214431 (2006)

    Article  ADS  Google Scholar 

  8. 8.

    Nomura, K.: Onset of incommensurability in quantum spin chain. J. Phys. Soc. Jpn. 72, 476–478 (2003)

    Article  ADS  Google Scholar 

  9. 9.

    Schollwöck, U., Jolicoeur, Th., Garel, Th.: On the onset of incommensurability at the VBS point in the S=1 bilinear-biquadratic quantum spin chain. Phys. Rev. B 53, 3304 (1996)

    Article  ADS  Google Scholar 

  10. 10.

    Yarotsky, D.: Ground states in relatively bounded quantum perturbations of classical lattice systems. Commun. Math. Phys. 261, 799–819 (2006)

    MATH  Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to D. A. Yarotsky.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yarotsky, D.A. Random Walk Analysis of the Commensurate-Incommensurate Transition in the Isotropic Spin-1 Chain. J Stat Phys 130, 957–981 (2008). https://doi.org/10.1007/s10955-007-9458-y

Download citation

Keywords

  • Commensurate-incommensurate transition
  • Isotropic spin-1 chain
  • AKLT model