Off-equilibrium scaling behaviors driven by time-dependent external fields in three-dimensional O(N) vector models

We consider the dynamical off-equilibrium behavior of the three-dimensional O(N) vector model in the presence of a slowly varying time-dependent spatially uniform magnetic field H(t)=h(t)e, where e is an N-dimensional constant unit vector, h(t)=t/ts, and ts is a time scale, at fixed temperature T≤Tc, where Tc corresponds to the continuous order-disorder transition. The dynamic evolutions start from equilibrium configurations at hi0 with h(tf)>0, or vice versa. We show that the magnetization displays an off-equilibrium scaling behavior close to the transition line H(t)=0. It arises from the interplay among the time t, the time scale ts, and the finite size L. The scaling behavior can be parametrized in terms of the scaling variables tsκ/L and t/tsκt, where κ>0 and κt>0 are appropriate universal exponents, which differ at the critical point and for T