Wavepackets in inhomogeneous periodic media: effective particle-field dynamics and Berry curvature
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.
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https://hdl.handle.net/10161/14060Published Version (Please cite this version)
10.1063/1.4976200Publication Info
Watson, AB; Lu, J; & Weinstein, MI (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.
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Show full item recordScholars@Duke
Jianfeng Lu
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 and other related fields.More specifically, his current research focuses include:Electronic
structure and many body problems; quantum molecular dynamics; multiscale modeling
and analysis; rare events and sampling techniques.
Alexander Watson
William W. Elliott Assistant Research Professor
I am an applied mathematician interested in partial differential equations, numerical
methods, and mathematical physics. My research has focused on problems arising in
condensed matter physics and photonics. My post-doc mentor at Duke is Jianfeng Lu.
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