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