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 ${\bm H}(t) = h(t)\,{\bm e}$, where ${\bm e}$ is a $N$-dimensional constant unit vector, $h(t)=t/t_s$, and $t_s$ is a time scale, at fixed temperature $T\le T_c$, 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 ${\bm H}(t)=0$. It arises from the interplay among the time $t$, the time scale $t_s$, 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.