Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields

Giuseppe de Nittis (1) , Max Lein (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Mathematisches Institut [Tübingen]

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 in [Pan07] to include periodic zero flux magnetic fields which is the setting also investigated in [Kuc09]. The new notion of magnetic symmetry plays a crucial role; 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<2m (stable rank regime) only the exponential localization of a subset of Wannier functions is shown; this improves part of the analysis of [Kuc09]. 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 trivility and thus exponential localization.

Mots clés: Wannier functions; magnetic symmetries; Bloch bundle

Domaines: Physique mathématique [math-ph]

Giuseppe de Nittis, Max Lein. Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields. Journal of Mathematical Physics, 2011, 52 (11), pp.112103. ⟨10.1063/1.3657344⟩. ⟨hal-00674265⟩