Math @ Duke

Publications [#328439] of Alexander Watson
Papers Published
 Watson, AB; Lu, J; Weinstein, MI, Wavepackets in inhomogeneous periodic media: Effective particlefield dynamics and Berry curvature,
Journal of Mathematical Physics, vol. 58 no. 2
(February, 2017),
pp. 021503021503, AIP Publishing [doi]
(last updated on 2020/07/31)
Abstract: © Published by AIP Publishing. We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger's equation with a potential which is the sum of a periodic function and a general smooth function. We identify two dimensionless parameters: (rescaled) 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 ∈. In the limit ∈ ↓ 0, we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to "Ehrenfest time" t ln 1/∈. To leading order, the center of mass and average quasimomentum 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 ∈. The corrections depend on the gaugeinvariant Berry curvature of the Bloch band and a coupling to the evolution of the wavepacket envelope, which satisfies Schrödinger's equation with a timedependent harmonic oscillator Hamiltonian. This infinite dimensional coupled "particlefield" system may be derived from an "extended" ∈dependent Hamiltonian. It is known that such coupling of observables (discrete particlelike degrees of freedom) to the waveenvelope (continuum fieldlike degrees of freedom) can have a significant impact on the overall dynamics.


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