Quantized magnetization density in periodically driven systems

We identify a new bulk quantized observable – the magnetization density -- that serves as a topological order parameter for periodically driven systems in which all bulk Floquet eigenstates are localized by disorder. While all Floquet states are localized when considered stroboscopically over a full period, the micromotion within the driving period may carry a nontrivial orbital magnetization. We find that the time-averaged magnetization density when the system is filled with fermions is quantized in units of the inverse driving period. We furthermore show that a quantized current flows around the boundary of any filled region of finite extent. The quantization has a topological origin: we relate the time-averaged magnetization density to the winding number characterizing the new phase identified in Phys. Rev. X 6, 021013 (2016). We thus establish that the winding number invariant can be accessed directly in bulk measurements, and propose an experimental protocol to do so using interferometry in a system of cold atoms in an optical lattice.