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 $N$). {\color{red}We discuss how this behaviour is a manifestation of an underlying dynamical scaling symmetry due to the presence of a single characteristic time $\tau _\alpha (N)\sim N^\alpha$.

中文翻译：

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远程Ising链中的粗化和平均场松弛并存

我们研究了一维Ising模型的低温淬灭后的动力学，其中自变量之间的远距离相互作用在距离$ r $处衰减为$ r ^ {-\ alpha} $。对于α= 0，即平均磁场，所有自旋连贯地发展，从而快速地将系统推向磁化状态。在$α> 1 $的弱远距离状态下，具有相反符号竞争域的粗化行为而不会磁化。对于强大的长距离，即$ 0 <\ alpha <1 $，我们表明系统显示了两个特征，其中概率$ P_ \ alpha（N）$具有后者，而具有不同的限制行为$ \ lim _ {N \ to \ infty} P_ \ alpha（N）= 0 $（固定为$ \ alpha <1 $）和$ \ lim _ {\ alpha \ to 1} P_ \ alpha（N）= 1 $（固定为固定$ N $）。