Effective Quantum Theories for Transport in Inhomogeneous Systems with   Non-trivial Band Structure
  Starting from a general $N$-band Hamiltonian with weak spatial and temporal variations, we derive a low energy effective theory for transport within one or several overlapping bands. To this end, we use the Wigner representation that allows us to systematically construct the unitary transformation that brings the Hamiltonian into band-diagonal form. We address the issue of gauge invariance and discuss the necessity of using kinetic variables in order to obtain a low energy effective description that is consistent with the original theory. Essentially, our analysis is a semiclassical one and quantum corrections appear as Berry curvatures in addition to quantities that are related to the appearance of persistent currents. We develop a transport framework which is manifestly gauge invariant and it is based on a quantum Boltzman formulation along with suitable definitions of current density operators such that Liouville's theorem is satisfied. Finally, we incorporate the effects of an external electromagnetic field into our theory. 
Christian Wickles and Wolfgang Belzig
cond-mat.mes-hall