Skip to main content
SpringerLink
Log in

Topological Invariants of Edge States for Periodic Two-Dimensional Models

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

Transfer matrix methods and intersection theory are used to calculate the bands of edge states for a wide class of periodic two-dimensional tight-binding models including a sublattice and spin degree of freedom. This allows to define topological invariants by considering the associated Bott–Maslov indices which can be easily calculated numerically. For time-reversal symmetric systems in the symplectic universality class this leads to a \({\mathbb Z}_2\)-invariant for the edge states. It is shown that the edge state invariants are related to Chern numbers of the bulk systems and also to (spin) edge currents, in the spirit of the theory of topological insulators.

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

Similar content being viewed by others

References

  1. Ando, T.: Numerical study of symmetry effects on localization in two dimensions. Phys. Rev. B40, 5325–5339 (1989)

    MathSciNet  ADS  Google Scholar 

  2. Elbau, P., Graf, G.-M.: Equality of bulk and edge Hall conductance revisited. Commun. Math. Phys. 229, 415–432 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Fröhlich, J., Studer, U.M., Thiran, E.: Quantum theory of large systems of non-relativistic matter. In: Les Houches Lectures 1994. Elsevier, New York (1996)

    Google Scholar 

  4. Fujita, M., Wakabayashi, K., Nakada, K., Kusakabe, K.: Peculiar localized state at zigzag graphite edge. J. Phys. Soc. Jpn. 65, 1920–1923 (1996)

    Article  ADS  Google Scholar 

  5. Graf, G.M., Porta, M.: Bulk-edge correspondence for two-dimensional topological insulators. arXiv:1207.5989

  6. Halperin, B.I.: Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185–2190 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  7. Hatsugai, Y.: The Chern number and edge states in the integer quantum hall effect. Phys. Rev. Lett. 71, 3697–3700 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Hatsugai, Y., Fukui, T., Aoki, H.: Topological analysis of the quantum Hall effect in graphene: Dirac-Fermi transition across van Hove singularities and edge versus bulk quantum numbers. Phys. Rev. B74, 205414–205430 (2006)

    ADS  Google Scholar 

  9. Kane, C.L., Mele, E.J.: \({\mathbb Z}_2\) topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802–145805 (2005)

    ADS  Google Scholar 

  10. Kellendonk, J., Richter, T., Schulz-Baldes, H.: Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14, 87–119 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Krein, M.G.: Principles of the theory of λ-zones of stability of a canonical system of linear differential equations with periodic coefficients. In: Memory of A.A. Andronov, pp. 413–498. Izdat. Akad. Nauk SSSR, Moscow (1955) (English Transl.: Krein, M.G.: Topics in Differential and Integral Equations and Operator Theory. Birkhäuser, Boston (1983))

  12. Kuchment, P.: Quantum Graphs II: some spectral properties of quantum and combinatorial graphs. J. Phys. A38, 4887–4900 (2005)

    MathSciNet  ADS  Google Scholar 

  13. Nishino, S., Goda, M., Kusakabe, K.: Flat bands of a tight-binding electronic system with hexagonal structure. J. Phys. Soc. Jpn. 72, 2015–2023 (2003)

    Article  ADS  Google Scholar 

  14. Prodan, E.: Robustness of the spin-Chern number. Phys. Rev. B80, 125327–125333 (2009)

    ADS  Google Scholar 

  15. Sadel, C., Schulz-Baldes, H.: Random Dirac operators with time reversal symmetry. Commun. Math. Phys. 295, 209–242 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Schulz-Baldes, H.: Rotation numbers for Jacobi matrices with matrix entries. Math. Phys. Electron. J. 13, 40 pp. (2007)

    MathSciNet  Google Scholar 

  17. Schulz-Baldes, H.: Geometry of Weyl theory for Jacobi matrices with matrix entries. J. Anal. Math. 110, 129–165 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schulz-Baldes, H., Kellendonk, J., Richter, T.: Edge versus Bulk currents in the integer quantum hall effect. J. Phys. A33, L27–L32 (2000)

    MathSciNet  ADS  Google Scholar 

  19. Sheng, D.N., Weng, Z.Y., Sheng, L., Haldane, F.D.M.: Quantum spin-hall effect and topologically invariant Chern numbers. Phys. Rev. Lett. 97, 036808–036811 (2006)

    Article  ADS  Google Scholar 

  20. Shockley, W.: On the surface states associated with a periodic potential. Phys. Rev. 56, 317–323 (1939)

    Article  ADS  MATH  Google Scholar 

  21. Tamm, I.: Über eine mögliche Art der Elektronenbindung an Kristalloberflächen. Z. Phys. Sov. 76, 849–850 (1932)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UNAM, Instituto de Matematicas, Cuernavaca, Mexico

    Julio Cesar Avila, Hermann Schulz-Baldes & Carlos Villegas-Blas

  2. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Hermann Schulz-Baldes

Authors
  1. Julio Cesar Avila
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hermann Schulz-Baldes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carlos Villegas-Blas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hermann Schulz-Baldes.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Avila, J.C., Schulz-Baldes, H. & Villegas-Blas, C. Topological Invariants of Edge States for Periodic Two-Dimensional Models. Math Phys Anal Geom 16, 137–170 (2013). https://doi.org/10.1007/s11040-012-9123-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11040-012-9123-9

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation