We study the quantum dynamics of a charged particle in a two-dimensional lattice, subject to constant and homogeneous electric and magnetic fields. We find that different regimes characterize these motions, depending on a combination of conditions, c
orresponding to weak and strong electric field intensities, rational or irrational directions of the electric field with respect to the lattice, and small or large values of the magnetic (Peierls) phase.
This paper studies the quantum dynamics of a charged particle in a 2D square lattice, under the influence of electric and magnetic fields, the former being aligned with one of the lattice axes and the latter perpendicular to the lattice plane. While
in free space these dynamics consist of uniform motions in the direction orthogonal to the electric field vector, we find that, in a lattice, this directed drift takes place only for specific initial conditions and for electric field magnitudes smaller than a critical value. Otherwise, the quantum wave--packet spreads ballistically in both directions orthogonal to the electric field. We quantify this ballistic spreading and identify the subspace of initial conditions insuring directed transport with the drift velocity. We also describe the effect of disorder in the system.
