Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature
Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature
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非均匀周期介质中的波包：有效粒子场动力学和贝里曲率
DOI：
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
M. Weinstein
中科院分区：
文献类型：
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作者：
Alexander B. Watson;Jianfeng Lu;M. Weinstein
我们考虑晶体中电子在外电场影响下运动的模型：薛定谔方程，其电位是周期函数和一般光滑函数的和。我们确定了两个无量纲参数：（重新缩放）普朗克常数和晶格间距与外部势变化尺度的比值。我们考虑两个参数相等的特殊情况，并将这个参数表示为ε。在极限λ↓0中，我们证明了被称为半经典波包的解的存在性，其渐近直至“Ehrenfest时间”t ~ ln1/ λ。在第一阶，这些解的质心和平均准动量沿着由布洛赫带能量和外部势的和给出的经典哈密顿量产生的轨迹演化。然后，我们推导出这些可观测值与λ成正比的所有修正。修正取决于布洛赫带的量规不变的贝里曲率和波包包络的演化耦合，这满足薛定谔方程与时间相关的谐振子哈密顿量。这个无限维耦合的“粒子场”系统可以从一个“扩展的”ϵ-dependent哈密顿量推导出来。众所周知，这种可观测值（离散类粒子自由度）与波包络（连续类场自由度）的耦合可以对整体动力学产生重大影响。我们考虑晶体中电子在外电场影响下运动的模型：薛定谔方程，其电位是周期函数和一般光滑函数的和。我们确定了两个无量纲参数：（重新缩放）普朗克常数和晶格间距与外部势变化尺度的比值。我们考虑两个参数相等的特殊情况，并将这个参数表示为ε。在极限λ↓0中，我们证明了被称为半经典波包的解的存在性，其渐近直至“Ehrenfest时间”t ~ ln1/ λ。在第一阶，这些解的质心和平均准动量沿着由布洛赫带能量和外部势的和给出的经典哈密顿量产生的轨迹演化。然后，我们推导出这些可观测值与λ成正比的所有修正。修正依赖于布洛赫带的量规不变的贝里曲率和与e…
We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrodinger’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 ϵ. In the limit ϵ↓0, we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to “Ehrenfest time” t∼ln1/ϵ. 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 ϵ. 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 Schrodinger’s equation with a time-dependent harmonic oscillator Hamiltonian. This infinite dimensional coupled “particle-field” system may be derived from an “extended” ϵ-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.We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrodinger’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 ϵ. In the limit ϵ↓0, we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to “Ehrenfest time” t∼ln1/ϵ. 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 ϵ. The corrections depend on the gauge-invariant Berry curvature of the Bloch band and a coupling to the e...
具有算子值符号的哈密顿量的半经典近似
影响因子：
2.4
作者：
Stiepan;Teufel
通讯作者：
Teufel