Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond

We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, (x), and vector potential A(x), with x ^d and 1. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of L ² ( ^d ) and an effective Hamiltonian governing the evolution inside this subspace to all orders in . 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.