Communications in Mathematical Physics

, Volume 359, Issue 1, pp 61–100 | Cite as

Optimal Decay of Wannier functions in Chern and Quantum Hall Insulators

  • Domenico Monaco
  • Gianluca PanatiEmail author
  • Adriano Pisante
  • Stefan Teufel


We investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e. g. in Chern insulators and in quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as \({|x| \rightarrow \infty}\), of the composite (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy in the following sense: either there exist exponentially localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari–Vanderbilt localization functional, is \({+ \infty}\). In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fachbereich Mathematik, Eberhard Karls Universität TübingenTübingenGermany
  2. 2.Dipartimento di Matematica“La Sapienza” Università di RomaRomeItaly

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