# Exponentially localized Wannier functions in periodic zero flux magnetic fields

de Nittis G, Lein M (2011)

**Publication Type:** Journal article

**Publication year:** 2011

### Journal

**Publisher:** American Institute of Physics (AIP)

**Book Volume:** 52

**Article Number:** 112103

**DOI:** 10.1063/1.3657344

### Abstract

In this work, we investigate conditions which ensure the existence of an exponentially localized Wannier basis for a given periodic hamiltonian. We extend previous results [Panati, G., Ann. Henri Poincare8, 995–1011 (2007)10.1007/s00023-007-0326-8] to include periodic zero flux magnetic fields which is the setting also investigated by Kuchment [J. Phys. A: Math. Theor.42, 025203 (2009)10.1088/1751-8113/42/2/025203]. The new notion of *magnetic symmetry* plays a crucial rôle; to a large class of symmetries for a non-magnetic system, one can associate “magnetic” symmetries of the related magnetic system. Observing that the existence of an exponentially localized Wannier basis is equivalent to the triviality of the so-called Bloch bundle, a rank *m* hermitian vector bundle over the Brillouin zone, we prove that *magnetic* time-reversal symmetry is sufficient to ensure the triviality of the Bloch bundle in spatial dimension*d* = 1, 2, 3. For *d* = 4, an exponentially localized Wannier basis exists provided that the *trace per unit volume* of a suitable function of the Fermi projection vanishes. For *d* > 4 and *d* ⩽ 2*m* (stable rank regime) only the exponential localization of a subset of Wannier functions is shown; this improves part of the analysis of Kuchment [J. Phys. A: Math. Theor.42, 025203 (2009)10.1088/1751-8113/42/2/025203]. Finally, for *d* > 4 and *d* > 2*m* (unstable rank regime) we show that the mere analysis of Chern classes does not suffice in order to prove triviality and thus exponential localization.

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### How to cite

**APA:**

de Nittis, G., & Lein, M. (2011). Exponentially localized Wannier functions in periodic zero flux magnetic fields. *Journal of Mathematical Physics*, *52*. https://doi.org/10.1063/1.3657344

**MLA:**

de Nittis, Giuseppe, and Max Lein. "Exponentially localized Wannier functions in periodic zero flux magnetic fields." *Journal of Mathematical Physics* 52 (2011).

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