Wavepackets in inhomogeneous periodic media: effective particle-field dynamics and Berry curvature

Loading...
Thumbnail Image

Date

2017-04-23

Journal Title

Journal ISSN

Volume Title

Repository Usage Stats

203
views
163
downloads

Citation Stats

Attention Stats

Abstract

We consider a model of an electron in a crystal moving under the influence of an external electric field: Schr"{o}dinger's equation with a potential which is the sum of a periodic function and a general smooth function. We identify two dimensionless parameters: (re-scaled) Planck's constant and the ratio of the lattice spacing to the scale of variation of the external potential. We consider the special case where both parameters are equal and denote this parameter $\epsilon$. In the limit $\epsilon \downarrow 0$, we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to Ehrenfest time' $t \sim \ln 1/\epsilon$. To leading order, the center of mass and average quasi-momentum of these solutions evolve along trajectories generated by the classical Hamiltonian given by the sum of the Bloch band energy and the external potential. We then derive all corrections to the evolution of these observables proportional to $\epsilon$. The corrections depend on the gauge-invariant Berry curvature of the Bloch band, and a coupling to the evolution of the wave-packet envelope which satisfies Schr\"{o}dinger's equation with a time-dependent harmonic oscillator Hamiltonian. This infinite dimensional coupled particle-field' system may be derived from an `extended' $\epsilon$-dependent Hamiltonian. It is known that such coupling of observables (discrete particle-like degrees of freedom) to the wave-envelope (continuum field-like degrees of freedom) can have a significant impact on the overall dynamics.

Department

Description

Provenance

Citation

Published Version (Please cite this version)

10.1063/1.4976200

Publication Info

Watson, AB, J Lu and MI Weinstein (2017). Wavepackets in inhomogeneous periodic media: effective particle-field dynamics and Berry curvature. 10.1063/1.4976200 Retrieved from https://hdl.handle.net/10161/14060.

This is constructed from limited available data and may be imprecise. To cite this article, please review & use the official citation provided by the journal.

Scholars@Duke

Lu

Jianfeng Lu

James B. Duke Distinguished Professor of Mathematics

Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm development for problems from computational physics, theoretical chemistry, materials science, machine learning, and other related fields.

More specifically, his current research focuses include:
High dimensional PDEs; generative models and sampling methods; control and reinforcement learning; electronic structure and many body problems; quantum molecular dynamics; multiscale modeling and analysis.


Unless otherwise indicated, scholarly articles published by Duke faculty members are made available here with a CC-BY-NC (Creative Commons Attribution Non-Commercial) license, as enabled by the Duke Open Access Policy. If you wish to use the materials in ways not already permitted under CC-BY-NC, please consult the copyright owner. Other materials are made available here through the author’s grant of a non-exclusive license to make their work openly accessible.