Thouless adiabatic quantum pumping is a cornerstone of condensed matter physics. It is a conceptual key for understanding many topological effects related to the topology of the bands and played a central role in the development of the modern theory of polarization. Recently, systems explicitly realizing a topological quantum pump have been realized in cold-atom experiments. 

In this talk, using Floquet theory, I will show some new results regarding the robustness of this phenomenon First, I will discuss what happens out of the perfect adiabatic limit. Our findings show that despite its topological nature, this phenomenon is not generically robust to non-adiabatic effects and depends crucially on the initial part of the driving protocol. 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 non-analytic behaviour on top of an overall quadratic decrease, in accordance with a theorem of Avron and Kons.

As a second point, I will discuss some preliminary results on the analysis of disorder effects. We prove that despite the localization of Hamiltonian eigenstates, quantized pumping in the adiabatic limit is stable after many cycles thanks to the existence of truly delocalized Floquet states. These states are  consequence of the topological nature of the driving and undergo a true localization phase transition upon increasing disorder, which is reflected in the breakdown of quantized transport.