Abstract
We study the semi-classical limit of the Schrödinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the asymptotic equations governing this transform in the semi-classical setting. For the second part, we focus on the appearance of the Berry curvature terms in the asymptotic equations. These terms play a crucial role in many important physical phenomena such as the quantum Hall effect. We give a simple derivation of these terms in different settings using asymptotic analysis.
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This work is supported in part by Department of Energy under Contract No. DE-FG02-03ER25587, by Office of Naval Research under Contract No. N00014-01-1-0674 and by National Science Foundation grant DMS-0708026.
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E, W., Lu, Jf. & Yang, X. Asymptotic analysis of quantum dynamics in crystals: the Bloch-Wigner transform, Bloch dynamics and Berry phase. Acta Math. Appl. Sin. Engl. Ser. 29, 465–476 (2013). https://doi.org/10.1007/s10255-011-0095-5
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DOI: https://doi.org/10.1007/s10255-011-0095-5