Off-equilibrium scaling behaviors driven by time-dependent external fields in three-dimensionalO(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 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 t(s), and the finite size L. The scaling behavior can be parametrized in terms of the scaling variables t(s)(κ)/L and t/t(s)(κ(t)), where κ>0 and κ(t)>0 are appropriate universal exponents, which differ at the critical point and for T<T(c). In the latter case, κ and κ(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.

Dynamical off-equilibrium scaling across magnetic first-order phase transitions

“…The scaling relations in eqs (4.5,4.7) apply beyond the spherical limit n → ∞ and are in agreement with the numerically obtained scaling behaviour for a 3D Heisenberg ferromagnet [75]. Indeed, in the case T < T c , we showed that the dynamics is independent on the number of…”

“…As mentioned, all results presented in this work are derived in the large-n limit. However, we argued that the dynamics of the system is not affected by the number of transverse components so that our results apply for any finite n ≥ 2, as confirmed by a comparison with numerical studies [75].…”

“…Remarkably, these results apply beyond the large-n limit with 2/3 ≈ 0.66, as can be seen by a comparison with numerical results by Pelissetto and Vicari [75]. To complete our analysis, we shall numerically investigate the dynamics in the large-n limit and explicitly test our scaling predictions.…”

“…As this field is driven across the critical value h c = 0, the magnetisation has to flip in order to align with the final magnetic field h > 0. Therefore, the vector of the magnetisation has to perform a rotation which needs a characteristic time of the order of the system volume L D [75]. For further details on how to determine z in the low temperature regime, we refer to appendix B.…”

“…which is a quantifier of the deviation from equilibrium during the process [75,83]: the larger the deviations from equilibrium are, the larger is the value of A, while for a system in equilibrium…”