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 dimensiond = 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 ⩽ 2m (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 > 2m (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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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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