Geometrical phase effects on the Wigner distribution of Bloch electrons
  We investigate the dynamics of Bloch electrons using a density operator method and connect this approach with previous theories based on wave packets. We study non-interacting systems with negligible disorder and strong spin-orbit interactions, which have been at the forefront of recent research on spin-related phenomena. We demonstrate that the requirement of gauge invariance results in a shift in the position at which the Wigner function of Bloch electrons is evaluated. The present formalism also yields the correction to the carrier velocity arising from the Berry phase. The gauge-dependent shift in carrier position and the Berry phase correction to the carrier velocity naturally appear in the charge and current density distributions. In the context of spin transport we show that the spin velocity may be defined in such a way as to enable spin dynamics to be treated on the same footing as charge dynamics. Aside from the gauge-dependent position shift we find additional, gauge-covariant multipole terms in the density distributions of spin, spin current and spin torque. 
Dimitrie Culcer, Qian Niu