Effective quantum theories for Bloch dynamics in inhomogeneous systems with nontrivial 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 Boltzmann 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.

Figure 1

Illustration of the band-diagonalization scheme: Bloch-sphere representation for a simple two-band model $\mathcal{H}=\vec{\xi }(\mathbf{r},\mathbf{p},t)·\vec{\sigma }={\xi }_x{\sigma }_x+{\xi }_y{\sigma }_y+{\xi }_z{\sigma }_z$ where the vector is rotated by $U_0(\mathbf{r},\mathbf{p},t)$ such that it aligns along the $z$ axis at each point in phase space.

Figure 2

Illustration of the bands described by the Hamiltonian $\mathcal{H}$ and $\Delta$ is a typical energy scale for interband distances. The projection operator on the $i$th band is denoted by ${\mathcal{P}}_i$. For low-energy processes around the Fermi energy $E_f$, only one or few overlapping bands are relevant and thus contribute to, e.g., transport properties. Corrections due to the influence of other bands enter as Berry curvatures and are $\propto 1/\Delta$.