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

Abstract

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 $\sigma \le 1$, 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 $\sigma \le 1$.

Accepted for publication in JSTAT, Special issue: New Trends in Nonequilibrium Statistical Mechanics: Classical and Quantum Systems (nesmcq18) (minor changes)

References in corpus (4)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Exact results for curvature-driven coarsening in two dimensions
• One dimensional phase-ordering in the Ising model with space decaying interactions
• Effective mobility and diffusivity in coarsening processes

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