Critical fluctuations of time-dependent magnetization in ordering processes near the disorder-induced critical point

The disorder-induced critical phenomena of a random field Ising model are numerically investigated by analyzing time-dependent magnetization in ordering processes. We find that the intensity of fluctuations of magnetization, χ(t), attains a maximum value at a time t = τ in a normal phase and that χ(τ ) and τ exhibit divergences near the disorder-induced critical point. Furthermore, spin configurations around the time τ are characterized by a length scale, which also exhibits a divergence near the critical point. We estimate the critical exponents that characterize these power-law divergences by using a finite-size scaling method. Introduction. – It has been known that earthquakes, acoustic emissions in a deformed complex material and Barkhausen noise exhibit distinctive power-law behaviors [1, 2]. All these systems possess some disorder in the system parameters and also they are driven by a slowly varying external field. As the simplest model of such systems, a random field Ising model (RFIM) [3] under a slowly varying magnetic field has been investigated in order to elucidate the essential mechanism of the phenomena. By performing numerical experiments of the RFIM under a slowly varying magnetic field, it was found that the size distribution of avalanches, each of which represents a spin flipping in a connected region, becomes a power-law function for some strength of the disorder [4]. Since this occurs due to the existence of disorder, such power-law behaviors are called disorder-induced critical phenomena. The power-law behaviors observed in previous extensive studies [5–9] are clear evidences for some criticality, but the characterization in terms of avalanches in the quasi-static operation appears to be technical. Obviously, the simplest quantity in the RFIM is magnetization. With regard to magnetization, it has been conjectured since the report in Ref. [6] that a discontinuous jump occurs at the disorder-induced critical point. Thus, fluctuations of magnetization in equilibrium states does not exhibit a power-law divergence. In this sense, the nature of disorder-induced critical phenomena is different from that of conventional critical phenomena such as a ferromagnet-paramagnet transition. This raises a naive question whether the disorder-induced critical phenomena are characterized in terms of fluctuations of magnetization. In this paper, we present a positive answer to this question. Our key idea is to notice the recent extensive studies for critical fluctuations near an ergodicity breaking transition in glassy systems [10–16]. Here, we review these studies briefly. As an example, let us consider behaviors near the transition point in colloidal suspensions. In this system, near the transition point, there exist long-range spatial correlations among movable particles during a time interval τ , which is chosen as a typical relaxation time. When we introduce an appropriate quantity Q(r, t) that indicates the occurrence of large particle displacement at the position r during the time interval t, a discontinuous jump occurs in Q(r, t → ∞) accompanied with critical fluctuations of Q(r, τ) at the ergodicity breaking transition. It should be noted that Q(r, t) characterizes the dynamical event intrinsic to glassy systems. These results motivates us to study the disorder-induced critical phenomena of the RFIM from the viewpoint of critical fluctuations of some dynamical events. In particular, we consider ordering processes of the magnetization from a special initial condition. We have numerically found that fluctuations of the time-dependent magnetization exhibit a critically divergent behavior near the disorder-induced critical point.