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Aging and coarsening in isolated quantum systems after a quench: exact results for the quantum $O(N)$ model with $N \to \infty$

The non-equilibrium dynamics of an isolated quantum system after a sudden quench to a dynamical critical point is expected to be characterized by scaling and universal exponents due to the absence of time scales. We explore these features for a quench of the parameters of a Hamiltonian with $O(N)$ symmetry, starting from a ground state in the disordered phase. In the limit of infinite $N$, the exponents and scaling forms of the relevant two-time correlation functions can be calculated exactly. Our analytical predictions are confirmed by the numerical solution of the corresponding equations. Moreover, we find that the same scaling functions, yet with different exponents, also describe the coarsening dynamics for quenches below the dynamical critical point.