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 Pα(N) of having the latter one, with the different limiting behaviours limN→∞ Pα(N) = 0 (at fixed α < 1) and limα→1 Pα(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 τα(N) ∼ Nα.