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/t(s), and t(s) is a time scale, at fixed temperature T <= T-c, where T-c corresponds to the continuous order-disorder transition. The dynamic evolutions start from equilibrium configurations at h(i) < 0, correspondingly t(i) < 0, and end at time t(f) > 0 with h(t(f)) > 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 t(s)(kappa)/L and t/t(s)(kappa t), where kappa > 0 and kappa(t) > 0 are appropriate universal exponents, which differ at the critical point and for T < T-c. In the latter case, kappa and kappa(t) 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 T-c. 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.