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Dynamical trapping near a quantum critical point

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We consider a closed system where the parameter controlling a quantum phase transition is promoted to a dynamical field interacting with the quantum critical theory. In the case that the field has an energy extensive in the volume we can treat its evolution classically. We find that the field can become trapped near the phase transition point due to its interactions with the degrees of freedom of the quantum critical theory. The trapping/untrapping transition can be understood using Kibble-Zurek scaling arguments. We check the general framework numerically in the particular case of the 1D transverse field Ising chain, where the transverse magnetic field is dynamical. This constitutes a dynamical mechanism for tuning a relevant parameter to zero through a non-equilibrium process.