TY - JOUR

T1 - Exponentially localized Wannier functions in periodic zero flux magnetic fields

AU - De Nittis, G.

AU - Lein, M.

N1 - Funding Information:
G.D. is supported by the grant ANR-08-BLAN-0261-01. M.L. would like to thank the German-Israeli Foundation for kind support. Furthermore, G.D. and M.L. thank the Erwin Schrödinger Institute for its kind hospitality during the final drafting of this manuscript. We are also grateful to C. Rojas Molina, S. Teufel, and P. Kuchment for their helpful comments.

PY - 2011/11/1

Y1 - 2011/11/1

N2 - 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 ≤ 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.

AB - 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 ≤ 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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U2 - 10.1063/1.3657344

DO - 10.1063/1.3657344

M3 - Article

AN - SCOPUS:82855169664

VL - 52

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 11

M1 - 112103

ER -