布洛赫电子的有效动力学：Peierls代换及以后
Effective dynamics for Bloch electrons: Peierls substitution and beyond
摘要：
考虑电子在周期性势场中运动，外加一个缓慢变化的外静电势$ϕ(\epsi x)$和矢量势$A(\epsi x)$，其中$x \in \R^d$和$\epsi \ll 1$。我们证明了与孤立的Bloch带族相关联的$L^2(\R^d)$存在一个几乎不变的子空间和一个控制该子空间内演化到$\epsi$中所有阶的有效哈密顿量.对于首阶，通过Peierls代换给出了有效哈密顿量.我们显式计算一阶修正。通过对这个有效量子哈密顿量的半经典分析，我们建立了对固体物理标准半经典模型的一级修正。
英文摘要：
We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, $ϕ(\epsi x)$, and vector potential $A(\epsi x)$, with $x \in \R^d$ and $\epsi \ll 1$. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of $L^2(\R^d)$ and an effective Hamiltonian governing the evolution inside this subspace to all orders in $\epsi$. To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics.
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