In recent years, the dynamical consequences of spontaneous symmetry breaking have been investigated: What is the fate of the order parameter when the system is driven away from equilibrium?
Mean-field analyses suggest that dynamical criticality sistematically appears. However, they rigorously describe unrealistic infinite-range of infinite-dimensional limits, where few collective macroscopic variables play a role and all the microscopic degrees of freedom, associated with spatial fluctuations, are frozen. It is a matter of principle to understand whether such dynamical criticality is robust to the inclusion of fluctuations (that are present even at zero temperature in quantum systems): will they be able to drive the system to thermal equilibrium, and hence to trivialize the dynamical critical phenomenon into a standard equilibrium transition? If so, the above dynamical criticality would just represent a mean-field artefact.
We address this problem by studying the off-equilibrium dynamics of an infinite-range quantum Ising model in a transverse field with an additional short-range interaction. We present a viable systematic approach to deal with the time-evolution that goes beyond mean-field, in terms of a time-dependent spin-wave theory. The results are quite surprising: throughout the two phases the dynamical behavior of the order parameter remains largely unaffected by the coupling to the extensive "bath" of spatial fluctuations modes. On the other hand, such microscopic fluctuations turn out to have a deep impact on the dynamical critical point, giving rise to a whole new region with chaotic features, characterized by an "unpredictable" asymptotic dynamical order for long times. The latter non-trivial phenomenon, fully confirmed by numerical simulations of the full many-body quantum evolution, is completely absent at mean-field level.