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, φ(εx), and vector potential A(εx), with x ∈ ℝ^d and ε ≪ 1. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of L²(ℝ^d) and an effective Hamiltonian governing the evolution inside this subspace to all orders in ε. 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.