@article {2003,
title = {Effective dynamics for Bloch electrons: Peierls substitution and beyond},
number = {arXiv.org;math-ph/0212041v2},
year = {2003},
publisher = {Springer},
abstract = {We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, $\\\\phi(\\\\epsi x)$, 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.},
url = {http://hdl.handle.net/1963/3040},
author = {Gianluca Panati and Herbert Spohn and Stefan Teufel}
}