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We study the time evolution of the mass gap of the O(N) non-linear sigmamodel in 2+1 dimensions due to a time-dependent coupling in the large-$N$limit. Using the Schwinger-Keldysh approach, we derive a set of equations atlarge $N$ which determine the time dependent gap in terms of the coupling.These equations lead to a criterion for the breakdown of adiabaticity for slowvariation of the coupling leading to a Kibble-Zurek scaling law. We describe aself-consistent numerical procedure to solve these large-$N$ equations andprovide explicit numerical solutions for a coupling which starts deep in thegapped phase at early times and approaches the zero temperature equilibriumcritical point $g_c$ in a linear fashion. We demonstrate that for such aprotocol there is a value of the coupling $g= g_c^{\rm dyn}> g_c$ where the gapfunction vanishes, possibly indicating a dynamical instability. We study thedependence of $g_c^{\rm dyn}$ on both the rate of change of the coupling andthe initial temperature. We also verify, by studying the evolution of the massgap subsequent to a sudden change in $g$, that the model does not displaythermalization within a finite time interval $t_0$ and discuss the implicationsof this observation for its conjectured gravitational dual as a higher spintheory in $AdS_4$.

Non-equilibrium Dynamics of O(N) Nonlinear Sigma models: a Large-N approach