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We consider a two-dimensional system initialized in a topologically trivialstate before its Hamiltonian is ramped through a phase transition into a Cherninsulator regime. This scenario is motivated by current experiments withultracold atomic gases aimed at realizing time-dependent dynamics intopological insulators. Our main findings are twofold. First, consideringcoherent dynamics, the non-equilibrium Hall response is found to approach atopologically quantized time averaged value in the limit of slow butnon-adiabatic parameter ramps, even though the Chern number of the stateremains trivial. Second, adding dephasing, the destruction of quantum coherenceis found to stabilize this Hall response, while the Chern number genericallybecomes undefined. We provide a geometric picture of this phenomenology interms of the time-dependent Berry curvature.

Dynamical Buildup of a Quantized Hall Response from Non-Topological States