Non-equilibrium Dynamics of O(N) Nonlinear Sigma models: a Large-N approach
Abstract
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 starts deep in the gapped phase at early times and approaches the zero temperature equilibrium critical point
g
c
in a linear fashion. We demonstrate that for such a protocol there is a value of the coupling
g=
g
dyn
c
>
g
c
where the gap function vanishes, possibly indicating a dynamical instability. We study the dependence of
g
dyn
c
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
t
0
and discuss the implications of this observation for its conjectured gravitational dual as a higher spin theory in
Ad
S
4
.