Comment on “ Berry phase correction to electron density in

The main result of Xiao et al. [ Phys. Rev. Lett. 95, 137204 (2005)] is shown to follow from Hamiltonian mechanics. In a recent paper on the semiclassical dynamics of a Bloch electron, Xiao, Shi and Niu [1] claim that, due to a Berry curvature term, Liouville’s theorem on the conservation of the phasespace volume element would be violated, a fact which would have been overlooked so far. Their main results would then be to redefine the phase-space density by including a pre-factor they obtain so as to compensate for the non-invariance. Our aim here is to point out that all the mystery disappears if we use Hamiltonian mechanics. Giving a Hamiltonian structure amounts indeed to giving a Hamiltonian and a Poisson-bracket which satisfies the Jacobi identity; the equations of motion read ξ̇ = {H; ξ}, ξ = (ki, x ) [2]. The e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]