Skip to main content

A Stochastic Geometric Approach to Quantum Spin Systems

  • Chapter
Book cover Probability and Phase Transition

Part of the book series: NATO ASI Series ((ASIC,volume 420))

Abstract

We review some stochastic geometric models that arise from the study of certain quantum spin systems. In these models the fundamental properties of the ground states or equilibrium states of the quantum systems can be given a simple stochastic geometric interpretation. One thus obtains a new class of challenging stochastic geometric problems.

Work supported in part by NSF Grant PHY 92-14654.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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. (1990). Exact results on the dimerization transition in SU(n) antiferromagnetic chains. Journal of Physics C: Condensed Matter 2, 405–415.

    Article  ADS  Google Scholar 

  2. Affleck, I. (1991). Quantum spin chains and the Haldane gap. Journal of Physics C: Condensed Matter 1, 3047–3072.

    Article  ADS  Google Scholar 

  3. Affleck, I. and Lieb, E. H. (1986). A proof of part of Haldane’s conjecture on quantum spin chains. Letters in Mathematical Physics 12, 57–69.

    Article  MathSciNet  ADS  Google Scholar 

  4. Aizemnan, M., Chayes, J. T., Chayes, L., and Newman, C. M. (1988). Discontinuity of the magnetization in one-dimensional 1/∣x - y∣2 Ising and Potts models. Journal of Statistical Physics 50, 1–40.

    Article  MathSciNet  ADS  Google Scholar 

  5. Aizenman, M., Klein, A., and Newman, C. Percolation methods for disordered quantum Ising models. Preprint.

    Google Scholar 

  6. Aizenman, M. and Nachtergaele, B. Geometric aspects of quantum spin states. Preprint, and in preparation.

    Google Scholar 

  7. Batchelor, M. T. and Barber, M. (1990). Spin-s quantum chains and Temperley-Lieb algebras. Journal of Physics A: Mathematical and General 23, L15-L21.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Burton, R. M. and Keane, M. (1991). Topological and metric properties of infinite clusters in stationary two-dimensional site percolation. Israel Journal of Mathematics 76, 299–316.

    Article  MathSciNet  MATH  Google Scholar 

  9. Dyson, F. J., Lieb, E. H., and Simon, B. (1978). Phase transitions in quantum spin systems with isotropic and non-isotropic interactions. Journal of Statistical Physics 18, 335–383.

    Article  MathSciNet  ADS  Google Scholar 

  10. Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random cluster model I. Physica 57, 536–564.

    Article  MathSciNet  ADS  Google Scholar 

  11. Fortuin, C. M., Kasteleyn, P. W., and Ginibre, L. (1971). Correlation inequalities on some partially ordered sets. Communications in Mathematical Physics 22, 89–103.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Gandolfi, A., Keane, M., and Russo, L. (1988). On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation. Annals of Probability 16, 1147–1157.

    Article  MathSciNet  MATH  Google Scholar 

  13. Ginibre, J. (1968). Reduced density matrices for the anisotropic Heisenberg model. Communications in Mathematical Physics 10, 140–154.

    Article  MathSciNet  ADS  Google Scholar 

  14. Grimmett, G. R. (1994). Percolative problems. In Probability and Phase Transition (G. Grimmett, ed.), Kluwer, Dordrecht, pp. 69–86, this volume.

    Google Scholar 

  15. Haldane, F. D. M. (1983). Continuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the 0(3) nonlinear sigma model. Physics Letters A 93, 464–468.

    MathSciNet  ADS  MATH  Google Scholar 

  16. Kennedy, T., Lieb, E. H., and Shastri, B. S. (1988). Existence of Néel order in some spin 1/2 Heisenberg antiferromagnets. Journal of Statistical Physics 53, 383–415.

    Article  MathSciNet  ADS  Google Scholar 

  17. Kennedy, T. and Tasaki, H. (1992). Hidden symmetry breaking and the Haldane phase in S=1 quantum spin chains. Communications in Mathematical Physics 147, 431–484.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Klein, A. (1993). Multiscale analysis in disordered systems. This volume.

    Google Scholar 

  19. Klümper, A. (1990). The spectra of q-state vertex models and related antiferromagnetic quantum spin chains. Journal of Physics A: Mathematical and General 23, 809–823.

    Article  MathSciNet  ADS  Google Scholar 

  20. Koma, T. and Tasaki, H. Symmetry breaking in Heisenberg antiferromagnets. Communications in Mathematical Physics, to appear.

    Google Scholar 

  21. Lieb, E., Schulz, T., and Mattis, D. (1961). Two soluble models of an antiferromagnetic chain. Annals of Physics (NY) 16, 407–466.

    Article  ADS  MATH  Google Scholar 

  22. Newman, C. M. (1994). Disordered Ising systems and random cluster representations. In Probability and Phase Transition (G. Grimmett, ed.), Kluwer, Dordrecht, pp. 247–260, this volume.

    Google Scholar 

  23. Tasaki, H. (1991). Quantunm liquid in antiferromagnetic chains: a stochastic geometric approach to the Haldane gap. Physical Review Letters 66, 798–801.

    Article  MathSciNet  ADS  Google Scholar 

  24. Thomas, L. E. (1980). Quantum Heisenberg ferromagnets and stochastic exclusion processes. Journal of Mathematical Physics 21, 1921–1924.

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Nachtergaele, B. (1994). A Stochastic Geometric Approach to Quantum Spin Systems. In: Grimmett, G. (eds) Probability and Phase Transition. NATO ASI Series, vol 420. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8326-8_14

Download citation

  • DOI: https://doi.org/10.1007/978-94-015-8326-8_14

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4370-2

  • Online ISBN: 978-94-015-8326-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics