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29 March 2024
 
  » arxiv » math-ph/0212041

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Effective dynamics for Bloch electrons: Peierls substitution and beyond
Gianluca Panati ; Herbert Spohn ; Stefan Teufel ;
Date 12 Dec 2002
Journal Comm. Math. Phys. 242, 547-578 (2003)
Subject Mathematical Physics MSC-class: 81Q15; 81Q20 | math-ph cond-mat math.MP
AbstractWe 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.
Source arXiv, math-ph/0212041
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