Nonadiabatic Breaking of Topological Pumping

We study Thouless pumping out of the adiabatic limit. Our findings show that despite its topological nature, this phenomenon is not generically robust to nonadiabatic effects. Indeed, we find that the Floquet diagonal ensemble value of the pumped charge shows a deviation from the topologically quantized limit which is quadratic in the driving frequency for a sudden switch on of the driving. This is reflected also in the charge pumped in a single period, which shows a nonanalytic behavior on top of an overall quadratic decrease. Exponentially small corrections are recovered only with a careful tailoring of the driving protocol. We also discuss thermal effects and the experimental feasibility of observing such a deviation.