# Order parameter dynamics of the non-linear sigma model in the large $N$ limit

## TL;DR

This paper investigates the non-equilibrium dynamics of the order parameter in the non-linear sigma model at large N, revealing phenomena like synchronization under periodic drive and effective temperature after ramps.

## Contribution

It introduces a numerical scheme for solving Keldysh saddle point equations and explores the order parameter dynamics under ramps and periodic drives in the large N limit.

## Key findings

- Order parameter dynamics exhibit synchronization with drive frequency.
- Steady state temperature depends on ramp time and amplitude.
- Transient dynamics are controllable via drive parameters.

## Abstract

We study non-equilibrium order parameter dynamics of the non-linear sigma model in the large $N$ limit, using Keldysh formalism. We provide a scheme for obtaining stable numerical solutions of the Keldysh saddle point equations, and use them to study the order parameter dynamics of the model either following a ramp, or in the presence of a periodic drive. We find that the transient dynamics of the order parameter in the presence of a periodic drive is controlled by the drive frequency displaying the phenomenon of synchronization. We also study the approach of the order parameter to its steady state value following a ramp and find out the effective temperature of the steady state. We chart out the steady state temperature of the ordered phase as a function of ramp time and amplitude, and discuss the relation of our results to experimentally realizable spin models.