Universality in the time correlations of the long-range 1d Ising model

The equilibrium and nonequilibrium properties of ferromagnetic systems may be affected by the long-range nature of the coupling interaction. Here we study the phase separation process of a one-dimensional Ising model in the presence of a power-law decaying coupling, J(r) = 1/r(1+sigma) with sigma > 0, and we focus on the two-time autocorrelation function C(t, t(w)) = < s(i)(t)s(i)(t(w))>. We find that it obeys the scaling form C(t, t(w)) = f (L(t(w))/L(t)), where L(t) is the typical domain size at time t, and where f (x) can only be of two types. For sigma > 1, when domain walls diffuse freely, f (x) falls in the nearest-neighbour (nn) universality class. Conversely, for sigma <= 1, when domain walls dynamics is driven, f (x) displays a new universal behavior. In particular, the so-called Fisher-Huse exponent, which characterizes the asymptotic behavior of f (x) similar or equal to x(-lambda) for x >> 1, is lambda = 1 in the nn universality class (sigma > 1) and lambda = 1/2 for sigma <= 1.