Effective dynamics for Bloch electrons: Peierls substitution and beyond

TL;DR

This paper develops an advanced effective Hamiltonian framework for Bloch electrons in periodic potentials with external fields, extending the Peierls substitution to all orders and refining semiclassical models.

Contribution

It proves the existence of an almost invariant subspace and derives an effective Hamiltonian to all orders in the small parameter, including explicit first-order corrections.

Findings

Effective Hamiltonian valid to all orders in b5.

First-order correction to Peierls substitution explicitly computed.

Refined semiclassical model for Bloch electrons established.

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.