Quantum Response at Finite Fields

We study the response of driven, time dependent, quantum system whose instantaneous Hamiltonian has gaps in its spectrum. We consider the question if the breakdown of quantum transport, and in particular, the breakdown of the Integer Quantum Hall conductance, is a phase transition. We prove that, under mild conditions, quantum response has an isolated essential singularity and a vanishing holomorphic piece at zero fields, and that there are no singularities at finite field. We also study the quantum response in Floquet states. We argue, and give numerical evidence, that Chern numbers for eigenvectors of finite dimensional Floquet operators are generically zero.