Effective dynamics for Bloch electrons: Peierls substitution and beyond

Details Effective dynamics for Bloch electrons: Peierls substitution and beyond Effective dynamics for Bloch electrons: Peierls substitution and beyond

2002-12-12

arxiv.org/abs/math-ph/0212041v2

Comm. Math. Phys. 242, 547-578 (2003)

Physics

Mathematical Physics

37 pages

Scientific paper

10.1007/s00220-003-0950-1

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.

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