Skip to main content
SpringerLink
Log in

Rearrangement of Energy Bands: Chern Numbers in the Presence of Cubic Symmetry

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Formation of energy bands in the system of rotation-vibration quantum states of molecules is described within semi-quantum models under the presence of a symmetry group characterizing the equilibrium molecular configuration. Effective rotation-vibration Hamiltonians are written in two-quantum state models with rotational variables treated as classical ones. Eigen-line bundles associated with eigenvalues of 2×2 Hermitian matrix defined on rotational classical phase space which is a two-dimensional sphere are characterized by the first Chern class. Explicit procedure for the calculation of Chern numbers are suggested and realized for a family of Hamiltonians depending on extra control parameters in the presence of symmetry. Effective Hamiltonians for two vibrational states transforming according to some representations of the cubic symmetry group are studied. Chern numbers are evaluated for respective model Hamiltonians. The iso-Chern diagrams are introduced which split the parameter space into regions with fixed Chern numbers.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Abramovici, G., Kalugin, P.: Clifford modules and symmetries of topological insulators. arXiv:1101.1054 (2011); IJGMMP 9, N 3 (2012)

  2. Arnold, V.I.: Remarks on eigenvalues and eigenvectors of Hermitian matrices. Berry phase, adiabatic connections and quantum Hall effect. Sel. Math. 1, 1–19 (1995)

    Article  MATH  Google Scholar 

  3. Avron, J.E., Sadun, L., Segert, J., Simon, B.: Chern numbers, quaternions, and Berry’s phases in Fermi systems. Commun. Math. Phys. 124, 595–627 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Faure, F.: Generic description of the degeneracies in Harper-like models. J. Phys. A, Math. Gen. 27, 7519–7532 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Faure, F., Zhilinskii, B.I.: Topological Chern indices in molecular spectra. Phys. Rev. Lett. 85, 960–963 (2000)

    Article  Google Scholar 

  6. Faure, F., Zhilinskii, B.I.: Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum. Lett. Math. Phys. 55, 239–247 (2001)

    Article  MathSciNet  Google Scholar 

  7. Faure, F., Zhilinskii, B.: Qualitative features of intra-molecular dynamics. What can be learned from symmetry and topology. Acta Appl. Math. 70, 265–282 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Faure, F., Zhilinskii, B.I.: Topologically coupled energy bands in molecules. Phys. Lett. A 302, 242–252 (2002)

    Article  MATH  Google Scholar 

  9. Iwai, T., Zhilinskii, B.: Energy bands: Chern numbers and symmetry. Ann. Phys. 326, 3013–3066 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kitaev, A.: Periodic table for topological insulators and superconductors. In: Lebedev, V., Feigel’man, M. (eds.) Advances in Theoretical Physics: Landau Memorial Conference, Chernogolovka, Russia, 22–26 June 2008. AIP Conf. Proc., vol. 1134, pp. 22–30 (2009)

    Google Scholar 

  11. Kohmoto, M.: Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160, 343–354 (1985)

    Article  MathSciNet  Google Scholar 

  12. Michel, L., Zhilinskii, B.: Symmetry, invariants, topology. I. Basic tools. Phys. Rep. 341, 11–84 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pavlov-Verevkin, V.B., Sadovskii, D.A., Zhilinskii, B.I.: On the dynamical meaning of diabolic points. Europhys. Lett. 6, 573–578 (1988)

    Article  Google Scholar 

  14. Sadovskii, D.A., Zhilinskii, B.I.: Qualitative analysis of vibration-rotation Hamiltonian for spherical top molecules. Mol. Phys. 65, 109–128 (1988)

    Article  Google Scholar 

  15. Sadovskii, D.A., Zhilinskii, B.I.: Group theoretical and topological analysis of localized vibration-rotation states. Phys. Rev. A 47, 2653–2671 (1993)

    Article  Google Scholar 

  16. Sadovskii, D.A., Zhilinskii, B.I.: Monodromy, diabolic points, and angular momentum coupling. Phys. Lett. A 256, 235–244 (1999)

    Article  MathSciNet  Google Scholar 

  17. Zhilinskii, B.I.: Symmetry, invariants, and topology in molecular models. Phys. Rep. 341, 85–171 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhilinskii, B.I., Brodersen, S.: The symmetry of the vibrational components in T d molecules. J. Mol. Spectrosc. 163, 326–338 (1994)

    Article  Google Scholar 

  19. Zirnbauer, M.R.: Symmetry classes. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) Random Matrix Theory. Oxford University Press, London (2010)

    Google Scholar 

Download references

Acknowledgements

This work was started during the visit of B.Zh. to Kyoto in September–December 2010. The financial support of this visit by Kyoto University is greatly acknowledged.

Author information

Authors and Affiliations

  1. Kyoto University, Kyoto, Japan

    T. Iwai

  2. Université du Littoral Côte d’Opale, 59140, Dunkerque, France

    B. Zhilinskii

Authors
  1. T. Iwai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Zhilinskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Zhilinskii.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Iwai, T., Zhilinskii, B. Rearrangement of Energy Bands: Chern Numbers in the Presence of Cubic Symmetry. Acta Appl Math 120, 153–175 (2012). https://doi.org/10.1007/s10440-012-9694-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-012-9694-2

Keywords

Search

Navigation