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Asymptotic analysis of quantum dynamics in crystals: the Bloch-Wigner transform, Bloch dynamics and Berry phase

  • Weinan E
  • Jian-feng Lu
  • Xu Yang
Article

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.

Keywords

semiclassical limit Bloch-Wigner transform Bloch dynamics Berry phase asymptotic analysis 

2000 MR Subject Classification

81Q20 81Q05 35Q40 

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Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  3. 3.School of Mathematical SciencesPeking UniversityBeijingChina

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