Skip to main content
SpringerLink
Log in

On the Construction of Wannier Functions in Topological Insulators: the 3D Case

  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for three-dimensional time-reversal symmetric topological insulators, both of bosonic and of fermionic type, so that the bases in question are also compatible with time-reversal symmetry. This problem is translated in the study (of independent interest) of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We identify three \(\mathbb {Z}_2\)-valued complete invariants for these homotopy classes. When these invariants vanish, we provide an algorithm which constructs a “multi-step logarithm” that is employed to continuously deform the given family into a constant one, identically equal to the identity matrix. This algorithm leads to a constructive procedure to produce the composite Wannier bases mentioned above.

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. Brouder, Ch., Panati, G., Calandra, M., Mourougane, Ch., Marzari, N.: Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007)

    Article  ADS  Google Scholar 

  2. Cancès, É., Levitt, A., Panati, G., Stoltz, G.: Robust determination of maximally localized Wannier functions. Phys. Rev. B 95, 075114 (2017)

    Article  ADS  Google Scholar 

  3. Cornean, H.D., Herbst, I., Nenciu, G.: On the construction of composite Wannier functions. Ann. Henri Poincaré 17, 3361–3398 (2016)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Cornean, H.D., Monaco, D., Teufel, S.: Wannier functions and \({\mathbb{Z}}_{2}\) invariants in time-reversal symmetric topological insulators. Rev. Math. Phys. 29, 1730001 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fiorenza, D., Monaco, D., Panati, G.: Construction of real-valued localized composite Wannier functions for insulators. Ann. Henri Poincaré 17, 63–97 (2016)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Fiorenza, D., Monaco, D., Panati, G.: \({\mathbb{Z}}_{2}\) invariants of topological insulators as geometric obstructions. Commun. Math. Phys. 343, 1115–1157 (2016)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Fu, L., Kane, C.L.: Time reversal polarization and a \({\mathbb{Z}}_{2}\) adiabatic spin pump. Phys. Rev. B 74, 195312 (2006)

    Article  ADS  Google Scholar 

  8. Fu, L., Kane, C.L., Mele, E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007)

    Article  ADS  Google Scholar 

  9. Graf, G.M., Porta, M.: Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324, 851–895 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et equation de Harper. In: Jensen, A., Holden, H. (eds.) Schrödinger Operators. No. 345 in Lecture Notes in Physics, pp. 118–197. Springer, Berlin (1989)

    Chapter  Google Scholar 

  11. Husemoller, D.: Fibre bundles, 3rd edn. No. 20 in Graduate Texts in Mathematics. Springer, New York (1994)

  12. Kuchment, P.: Floquet Theory for Partial Differential Equations, 3rd edn. No. 60 in Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (1993)

  13. Kuchment, P.: Tight frames of exponentially decaying Wannier functions. J. Phys. A Math. Theor. 42, 025203 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Marzari, N., Mostofi, A.A., Yates, J.R., Souza, I., Vanderbilt, D.: Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419 (2012)

    Article  ADS  Google Scholar 

  15. Monaco, D.: Chern and Fu–Kane–Mele invariants as topological obstructions. In: Dell’Antonio, G., Michelangeli, A. (eds.) Advances in Quantum Mechanics: Contemporary Trends and Open Problems, pp. 201–222. Springer, Cham (2017). vol. 18 in Springer INdAM Series

    Chapter  Google Scholar 

  16. Monaco, D., Panati, G.: Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry. Acta Appl. Math. 137, 185–203 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  17. Nenciu, G.: Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians. Rev. Mod. Phys. 63, 91 (1991)

    Article  ADS  Google Scholar 

  18. Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. Henri Poincaré 8, 995–1011 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Soluyanov, A.A., Vanderbilt, D.: Smooth gauge for topological insulators. Phys. Rev. B 85, 115415 (2012)

    Article  ADS  Google Scholar 

  20. Wegge-Olsen, N.E.: K-Theory and \(C^*\)-Algebras. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  21. Winkler, G.W., Soluyanov, A.A., Troyer, M.: Smooth gauge and Wannier functions for topological band structures in arbitrary dimensions. Phys. Rev. B 93, 035453 (2016)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220, Aalborg, Denmark

    Horia D. Cornean

  2. Fachbereich Mathematik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Domenico Monaco

Authors
  1. Horia D. Cornean
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Domenico Monaco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Horia D. Cornean.

Additional information

Communicated by Vieri Mastropietro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cornean, H.D., Monaco, D. On the Construction of Wannier Functions in Topological Insulators: the 3D Case. Ann. Henri Poincaré 18, 3863–3902 (2017). https://doi.org/10.1007/s00023-017-0621-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-017-0621-y

Search

Navigation