Skip to main content
SpringerLink
Log in

Hamiltonian Monodromy as Lattice Defect

  • Chapter
Topology in Condensed Matter

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 150))

Summary

The analogy between monodromy in dynamical (Hamiltonian) systems and defect in crystal lattices is used in order to formulate some general conjectures about possible types of qualitative features of quantum systems that can be interpreted as a manifestation of classical monodromy in quantum finite particle (molecular) problems.

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. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1981)

    MATH  Google Scholar 

  2. Lerman, L.M., Umanskii, Ya.L.: Four dimensional integrable Hamiltonian systems with simple singular points, Translations of Mathematical Monographs, 176, AMS, Providence, RI (1998)

    Google Scholar 

  3. Zung, N.T.: Diff. Geom. Appl. 7, 123 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cushman, R.H.: C.W.I. Newslett. 1, 4 (1983)

    MathSciNet  Google Scholar 

  5. Duistermaat, J.J.: Comm. Pure Appl. Math. 33, 687 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cushman, R.H., Duistermaat, J.J.: Bull. Am. Math. Soc. 19, 475 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bates, L.: J. Appl. Math. Phys. (ZAMP) 42, 837 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Cushman, R.H., Bates, L.M.: Global Aspects of Classical Integrable Systems, Birkhäuser, Basel (1997)

    MATH  Google Scholar 

  9. Sadovskii, D.A., Zhilinskii, B.I.: Phys. Lett. A 256, 235 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  10. Nekhoroshev, N.N., Sadovskii, D.A., Zhilinskii, B.I.: C. R. Acad. Sci. Paris, Ser. I, 335, 985 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Colin de Verdier, Y., Vu Ngoc, S.: Ann. Ec. Norm. Sup. 36, 1 (2003)

    Google Scholar 

  12. Nekhoroshev, N.N., Sadovskii, D.A., Zhilinskii, B.I.: in preparation (2003)

    Google Scholar 

  13. Nekhoroshev, N.N.: Trans. Moscow Math. Soc. 26, 180 (1972)

    Google Scholar 

  14. Grondin, L., Sadovskií, D.A., Zhilinskii, B.I.: Phys. Rev. A, 142, 012105 (2002)

    ADS  Google Scholar 

  15. Vu Ngoc, S.: Comm. Math. Phys. 203, 465 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Waalkens, H., Dullin, H.R.: Ann. Phys. (NY) 295, 81 (2001)

    ADS  MathSciNet  Google Scholar 

  17. Child, M.S., Weston, T., Tennyson, J.: Mol. Phys. 96, 371 (1999)

    Article  ADS  Google Scholar 

  18. Kleman, M.: Points Lines and Walls. Chichester, Wiley (1983)

    Google Scholar 

  19. Mermin, N.D.: Rev. Mod. Phys. 51, 591 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  20. Michel, L.: Rev. Mod. Phys. 52, 617 (1980)

    Article  ADS  Google Scholar 

  21. Kroner, E.: Continuum Theory of Defects in Physics of Defects. Les Houches, Ecole d’ete de physique theorique. Elsevier, New York, 215–315 (1981)

    Google Scholar 

  22. Cushman, R., Vu Ngoc, S.: Ann. I. Henri Poincaré, 3, 883 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cushman, R.H., Sadovskií, D.A.: Physica D, 65, 166 (2000)

    Article  ADS  Google Scholar 

  24. Matveev, V.: Sb. Math. 187, 495 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  25. Cushman, R.H., Duistermaat, J.J.: J. Diff. Eqns. 172, 42 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Arnol’d, V., Avez, A.: Ergodic Problems of Classical Mechanics Benjamin, New York (1968)

    Google Scholar 

  27. Cushman, R., Zhilinskii, B.: J. Phys. A: Math. Gen 35, L415 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Bender, C., Boettcher, S.: Phys. Rev. Lett. 80, 5243 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Zhilinskii, B.: in Proceedings of 24th International Conference on Group-Theoretical Methods in Physics, Paris (2003)

    Google Scholar 

  30. Matsumoto, Y.: J. Math. Soc. Japan 37, 605 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  31. Cartan, E.: Leçon sur la Géométrie des Espaces de Riemann, Ch. 3. Gauthier-Villars, Paris (1928)

    Google Scholar 

  32. Zung, N.T.: ArXiv:math.DG/0010181 (2001)

    Google Scholar 

  33. Nelson, D.R.: Phys. Rev. B 28, 5515 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  34. Duistermaat, J.J., Heckman, G.J.: Invent. Math. 69, 259 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Guillemin, V.: Moment Map and Combinatorial Invariants of Hamiltonian T n-spaces. Birkhäuser, Basel (1994)

    Google Scholar 

  36. Bowick, M.J., Travesset, A.: Phys. Rep. 344, 255 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Krishnan, A. et al.: Nature 388, 451 (1997)

    Article  ADS  Google Scholar 

  38. Ganser, B.K. et al.: Science 283, 80 (1999)

    Article  ADS  Google Scholar 

  39. Gido, S.P., Gunther, J., Thomas, E.L., Hoffman, D.: Macromolecules 26, 4506 (1993)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université du Littoral, UMR du CNRS 8101, 59140, Dunkerque, France

    B. Zhilinskii

Authors
  1. B. Zhilinskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117259, Moscow, Russia

    Michail Ilych Monastyrsky

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Zhilinskii, B. (2006). Hamiltonian Monodromy as Lattice Defect. In: Monastyrsky, M.I. (eds) Topology in Condensed Matter. Springer Series in Solid-State Sciences, vol 150. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31264-1_8

Download citation

Publish with us

Policies and ethics

Search

Navigation