Absence of disordered Thouless pumps at finite frequency

We study the interplay of disorder and finite frequency driving in the Thouless pumps and show that the pump rate always decays to zero due to non-adiabatic transitions between the instantaneous eigenstates. However, the decay is slow, occurring on a time-scale that is exponentially large in the period of the drive. In the adiabatic limit, the band gap in the instantaneous spectrum closes at a critical disorder strength, above which pumping ceases. We predict the scaling of the pump rate around this transition from a model of scattering between rare states near the band edges.