Superaging in two-dimensional random ferromagnets

Abstract

We study the aging properties, in particular the two-time autocorrelations, of the two-dimensional randomly diluted Ising ferromagnet below the critical temperature via Monte-Carlo simulations. We find that the autocorrelation function displays additive aging C(t, tw) = Cst(t) + Cag(t, tw), where the stationary part Cst decays algebraically. The aging part shows anomalous scaling Cag(t, tw) = C(h(t)/h(tw)), where h(u) is a non-homogeneous function excluding a t/tw scaling. PACS numbers:The phase ordering kinetics in pure systems has attracted much attention in the last years [1]. A common scenario for instance for ferromagnets after a fast quench from above to below the ordering temperature is a continuous domain growth governed by a single length scale that depends algebraically on the time t w after the quench. The existence of this length scale quite frequently determines also the scaling properties of other dynamical non-equilibrium quantities like the two-time auto-correlation function C(t, t w ), which describes the correlations between the spin configurations at the time t w after the quench and a later configuration at a time t + t w . It gives rise to what is called simple aging in the context of glassy systems [2]: C(t, t w ) depends for large times t and t w only on the scaling variable t/t w . This behavior is rather well established by analytical works in various non-random models [3], and it has been corroborated by a large amount of numerical work [2].Much less analytical results are available for disordered ferromagnets, where numerical simulations thus play an important role. A recent numerical study of the relaxational dynamics in two-dimensional random magnets [4] found evidence for a power law growth L(t) ∝ t 1/z of the aforementioned length scale L(t). The dynamical exponent z turned out to depend both on temperature T and disorder strength and to behave as z ∝ 1/T at low temperatures T . The latter is compatible with activated dynamics of pinned domain walls over logarithmic free energy barriers (rather than power law [5]). The apparent existence of a single length scale that grows algebraically was confirmed by a recent numerical work [6], where it was furthermore claimed that the response function is well described by local scale invariance [7]. In spite of this, the correlation function showed systematic deviations from a simple t/t w scaling [6] (although simple aging seems to work well in d = 3 [8]). In [6], the autocorrelation was then compared to the scaling form C(t, t w ) ∼ t −xc (t/t w ), which usually holds at a critical point with x > 0 [9]. However, a fit of this form to the numerical data obtained in [6] (and to ours as we will report below) yielded negative exponents x, which is unphysical. The aim of this paper is to suggest an alternative picture originally applied in the context of aging experiments in glasses [10] and spin glasses [11].We study the site diluted Ising model (DIM) defined on 2-dimensional square lattice with periodic boundary conditions, and described by th...

Aging in coarsening diluted ferromagnets

“…One could try to bring the data to a collapse by letting b to differ from 0, but this yields the unphysical result that b Ͻ 0, implying that C would grow without bounds when s → ϱ. 12 This observation led Paul et al 12 to fit their numerical data to the superaging scaling ansatz ͓Eq. ͑2͔͒.…”

“…This is a rather hypothetical scenario as Kurchan has proven an exact lemma 25 that states that under very general conditions a superaging behavior of the autocorrelation cannot exist ͑this prove can also be extended to response functions 18 ͒. It is therefore intriguing that a study of the autocorrelation in the two-dimensional random-site Ising model 12 yielded data to which one can fit the superaging scaling form…”

“…Our simulations reveal that for this system the apparent power-law growth of the dynamical correlation length L͑t͒ only holds in the early stages of the coarsening process and that notable deviations are already encountered for moderate values of the correlation length. As a consequence one cannot naively assume that L͑t͒ϳt 1/z , as done in previous studies, 11,12 but instead the correct time dependence L͑t͒ has to be used in the investigation of the scaling behavior of two-time quantities. When doing this, a simple aging behavior emerges for the autocorrelation function in the scaling limit, thereby showing that the previously proposed superaging scenario is not appropriate for the description of aging in diluted ferromagnets relaxing toward equilibrium.…”

“…Monte Carlo simulations of various disordered ferromagnetic Ising models [10][11][12][13][14][15] found for the dynamical correlation length L͑t͒ a power-law increase, L͑t͒ϳt 1/z , with a nonuniversal dynamical exponent z that depends on temperature and on the nature of the disorder. This behavior can be explained by assuming that the energy barriers grow logarithmically with L. 10,11 As a consequence, it follows that the dynamical exponent z should be inversely proportional to the temperature: z ϳ 1 / T. Whereas this temperature dependence has been observed in the CardyOstlund model, 19 a careful study 14 has revealed that in the random-bond Ising model the temperature dependence of the dynamical exponent is more complicated.…”

“…One issue that has attracted much attention recently concerns ordering processes, relaxation phenomena, and the formation of domains in disordered systems. Examples include vortex lines in disordered type-II superconductors, [1][2][3][4][5] polymers in random media, [6][7][8][9] coarsening of disordered magnets, [10][11][12][13][14][15] and slow dynamics in spin glasses. 13,16,17 Our understanding of relaxation phenomena and aging processes in nonfrustrated systems with slow dynamics has greatly progressed during the last decade, mainly through the systematic study of critical and coarsening ferromagnets ͑see Ref.…”