Effect of an electric field on a split Bloch band

The effect of a constant and uniform electric field on a split Bloch band is studied through a tight binding model. The Wannier-Stark ladders of the parent band are modified by a potential in addition to that of the crystal. It is shown how the frequency spectrum behaves as the field strength varies from weak to strong. In the case that the parent band is split into a finite number of sub-bands, there exists a set of subspaces to which the states return periodically in time, and that the constituent states of each subspace are localized around a given unit cell. A theoretical framework is developed which naturally exploits the symmetries of the system, and which allows for easy and clean numerical calculations.