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 $C(t,t_w)=\langle s_i(t) s_i(t_w)\rangle$. 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 $\sigma \le 1$, 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 $f(x)\simeq x^{-\lambda}$ for $x\gg 1$, is $\lambda=1$ in the nn universality class ($\sigma > 1$) and $\lambda=1/2$ for $\sigma \le 1$.