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Half-integer quantized topological response in quasiperiodically driven quantum systems

A spin strongly driven by two harmonic incommensurate drives can pump energy from one drive to the other at a quantized average rate, in close analogy with the quantum Hall effect. The pumping rate is a non-zero integer in the topological regime, while the trivial regime does not pump. The dynamical transition between the regimes is sharp in the zero-frequency limit and is characterized by a Dirac point in a synthetic band structure. We show that the pumping rate is {\em half-integer} quantized at the transition and present universal Kibble-Zurek scaling functions for energy transfer processes. Our results adapt ideas from quantum phase transitions, quantum information and topological band theory to non-equilibrium dynamics, and identify qubit experiments to observe the universal linear and non-linear response of a Dirac point in synthetic dimensions.