Coexistence of coarsening and mean field relaxation in the long-range Ising chain

Abstract

We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance $r$ decaying as $r^{-\hat{I}\pm }$. For $\hat{I}\pm =0$, i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with $\hat{I}\pm >1$ there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. $0<\hat{I}\pm <1$, we show that the system shows both features, with probability $P_{\hat{I}}\pm (N)$ of having the latter one, with the different limiting behaviours ${\lim }_{N\to \infty }{P}_{\hat{I}}\pm (N)=0$ (at fixed $\hat{I}\pm <1$) and ${\lim }_{\hat{I}\pm \to 1}{P}_{\hat{I}}\pm (N)=1$ (at fixed finite $N$). We discuss how this behaviour is a manifestation of an underlying dynamical scaling symmetry due to the presence of a single characteristic time $\ddot{I}{„}_{\hat{I}}\pm (N)\sim N^{\hat{I}}\pm$.

11 pages, 3 figures