Non-equilibrium dynamics of O(N) nonlinear sigma models: A large-N approach

We study the time evolution of the mass gap of the O(N) non-linear sigma model 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 at large N which determine the time-dependent gap in terms of the coupling. These equations lead to a criterion for the breakdown of adiabaticity for slow variation of the coupling leading to a Kibble-Zurek scaling law. We describe a self-consistent numerical procedure to solve these large-N equations and provide explicit numerical solutions for a coupling which asymptotes to constant values in the gapped phase and approaches the zero temperature equilibrium critical point in a linear fashion. We demonstrate that for such a protocol there is a value of the coupling g = g_c^dyn > g_c where the gap function vanishes, possibly indicating a dynamical instability. We study the dependence of g_c^dyn on both the rate of change of the coupling and the initial temperature. We also verify, by studying the evolution of the mass gap subsequent to a sudden change in g, that the model does not display thermalization within a finite time interval t₀ and discuss the implications of this observation for its conjectured gravitational dual as a higher spin theory in AdS₄.