Abstract
In the previous chapter, we described quantized adiabatic pumping of particles in a one-dimensional lattice in an intuitive and visual fashion, using the concepts of the control-freak pumping cycle and the time evolution of the Wannier centers. Here, we provide a more formal description of the same effect. Based on Ehrenfest’s theorem, we identify the current operator describing the flow of probability density through a cross section of the one-dimensional lattice, and find that the momentum- and time-resolved current in a given filled band of the lattice is proportional to the Berry curvature associated to that band. Naturally, this leads to the same conclusion as we have seen before: that the number of particles adiabatically pumped through a cross section of the crystal is given by the Chern number of the corresponding filled band, and therefore it is an integer.
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