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

Watson, AB

Lu, J

Weinstein, MI

2017-04-23T15:48:23Z

2017-04-23T15:48:23Z

2017-04-23

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.

51 pages

http://arxiv.org/abs/1611.03408v2

https://hdl.handle.net/10161/14060

AIP Publishing

10.1063/1.4976200

math-ph

math-ph

cond-mat.mes-hall

math.AP

math.MP

quant-ph

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

Journal article

Lu, J|0000-0001-6255-5165

http://arxiv.org/abs/1611.03408v2

The following changes have been made: (1) revised introduction, (2) Section 1.5 of the earlier version has been removed, and replaced with a Remark 1.10 (in the current Section 1.4) and reference to T. Ohsawa & M. Leok (reference [29]). (3) Appendix D on certain uniform bounds in the quasi-momentum parameter, p, has been added. (4) Additional references have also been given

Chemistry

Duke

Mathematics

Physics

Trinity College of Arts & Sciences

http://dx.doi.org/10.1063/1.4976200