Critical aging of a ferromagnetic system from a completely ordered state

Abstract

We adapt the non-linear σ model to study the nonequilibrium critical dynamics of O(n) symmetric ferromagnetic system. Using the renormalization group analysis in d = 2 + ε dimensions we investigate the pure relaxation of the system starting from a completely ordered state. We find that the average magnetization obeys the long-time scaling behavior almost immediately after the system starts to evolve while the correlation and response functions demonstrate scaling behavior which is typical for aging phenomena. The corresponding fluctuation-dissipation ratio is computed to first order in ε and the relation between transverse and longitudinal fluctuations is discussed. [4] as in a ferromagnetic system below the critical temperature T c or by critical slowing down as in a ferromagnetic system exactly at criticality [5]. One expects that the critical aging phenomena can be cast into different universality classes of nonequilibrium critical dynamics. Most of studies consider the relaxation of a ferromagnetic system starting from a completely disordered state after quenching it to the fixed temperature T ≤ T c . It was found that the response function R(t, s) and the correlation function C(t, s) depend nontrivially on the ratio x = t/s similar to that found in glassy systems. Here s and t are waiting and observation times respectively. The distance from equilibrium can be measured by the fluctuation-dissipation ratio (FDR) X(t, s) = T R(t, s)/∂ s C(t, s). It has been argued that for critical aging the limit X ∞ = lim t,s→∞ X(t, s) is a novel universal quantity of critical phenomena [5]. The FDR was computed for the d dimensional spherical model [5], O(n) symmetric ferromagnetic model [6] and diluted spin models [7,8,9]. In all these systems X ∞ has values ranging between 0 and 1/2. The mean field calculations show that the aging behavior is modified in the presence of long-range correlations in the initial disordered state [10,11]. Much less known about the relaxation starting from an ordered state. The numerical simulations show that the correlation function demonstrates behavior which is typical for aging phenomena [8,12], while the magnetization obeys the long-time scaling behavior almost immediately after the system starts to evolve [13,14]. This observation was used to develop new effective numerical methods to determine the critical exponents, which are based on the short-time critical dynamics and do not require the time-consuming equilibration of the system [15]. However, up to now there is no any theoretical explanation why the long-time scaling behavior emerges already in the macroscopically early initial stage of relaxation and there is no theoretical framework which allows one to take properly into account the critical fluctuations in this regime of aging.In this paper we study the nonequilibrium critical dynamics of a ferromagnetic system starting from a completely ordered state. The long-distance properties of the O(n) symmetric system below the transition point can be related to the non-linear σ model d...

Critical ageing of Ising ferromagnets relaxing from an ordered state

“…We also compare the provided predictions with the results of Monte Carlo simulations of the two-dimensional Ising model with Glauber dynamics. The same problem has been recently addressed for other universality classes [30,9,31]. The comparison of our findings with these recent results is presented in the conclusive section.…”

Relaxation phenomena at criticality

“…1). An analogous conclusion has been drawn for different universality classes [14,15,18]. The full scaling functions in equation (4) actually describe the crossover which occurs, for a finite and fixed value of the magnetization M 0 (and therefore t 0 ) when both times s and t increases from the case s < t t 0 (described by Eq.…”

On the identification of quasiprimary scaling operators in local scale-invariance

“…It is remarkable that the two-time autocorrelations and autoresponses of models as physically different as those included in table 1 (and several further ones with a = a ′ which we did not include) can be described in terms of a single theoretical idea. On the other hand, field-theoretical studies of the critical O(n) model in both 4 − ε dimensions [5,29] and in 2 + ε dimensions [11], although they agree with LSI at the lowest orders in ε, continue to find discrepancies with either (8) or (9) at some higher order. However, non-equilibrium field-theory presently only yields explicit results for the first few terms of the ε-expansion series.…”

“…Until recently, the scaling (2) has only been studied for systems with a fully disordered initial state with mean initial magnetization m 0 = φ(0, r) = 0. The study of the effects of a non-vanishing initial magnetization on the ageing behaviour is only beginning [9,10,11]. We stress that in the kind of system under consideration invariance under time-translations is broken.…”

“…The scaling of two-time observables has been recently discussed in [9,10,11] and it was shown that for m 0 = 0 the universal scaling behaviour is different from the one found for m 0 = 0. An extension of LSI to non-equilibrium critical dynamics with non-vanishing initial magnetizations is an open problem to which to hope to return elsewhere.…”