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 hi<0, correspondingly ti<0, and end at time tf>0 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<Tc. In the latter case, κ and κt also depend on the shape of the lattice and on the boundary conditions. We present numerical results for the Heisenberg (N=3) model under a purely relaxational dynamics. They confirm the predicted off-equilibrium scaling behaviors at and below Tc. We also discuss hysteresis phenomena in round-trip protocols for the time dependence of the external field. We define a scaling function for the hysteresis loop area of the magnetization that can be used to quantify how far the system is from equilibrium.