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

Abstract

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 $\mathbf{H}(t)=h(t)\mathbf{e}$, where $\mathbf{e}$ 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 $\mathbf{H}(t)=0$. 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.

16 pages, extended text and refs

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