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We investigate the off-equilibrium dynamics of a spin system with O($N$)symmetry in $2 < d < 4$ spatial dimensions arising by the presence of a slowlyvarying time-dependent magnetic field $h(t,t_s) \sim t/t_s$, $t_s$ is a timescale, at the critical temperature $T = T_c$ and below it $T < T_c$. Aftershowing the general theory, we demonstrate the off-equilibrium scaling and weformally compute the correlation functions in the limit of large $N$. We derivethe off-equilibrium scaling relations for the hysteresis loop area and for themagnetic work done by the system when the magnetic field $h(t,t_s)$ is variedacross the phase transitions cyclically in time. We also investigate the firstdeviations from the equilibrium behavior in the correlation functions checkingthe consistence for an exponential approach.

Off-equilibrium scaling driven by a time-dependent magnetic field in O(N) vector models