Centralized distribution of adiabatic current in a split Bloch band

It is known that for a filled Bloch band, the particle transport in a cycle induced by a slow change of the periodic potential is quantized as integers. In this paper, we consider the case where a Bloch band is split by an extra static potential into a number of minibands. We study how the number of particle transports is distributed among the minibands, and when the distribution is concentrated at a few central minibands. Particular attention is given to situations where the splitting potential has a long spatial period and is random within a unit cell