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

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