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 $r^{-\alpha }$. For $\alpha =0$, i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with $\alpha >1$ there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. $0<\alpha <1$, we show that the system shows both features, with probability $P_{\alpha }(N)$ of having the latter one, with the different limiting behaviours ${\lim }_{N\to \infty }{P}_{\alpha }(N)=0$ (at fixed $\alpha <1$) and ${\lim }_{\alpha \to 1}{P}_{\alpha }(N)=1$ (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 ${\tau }_{\alpha }(N)\sim N^{\alpha }$.

arXiv comment11 pages, 3 figures

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