We study the kinetics after a low temperature quench of the
one-dimensional Ising model with long range interactions between spins
at distance rr
decaying as r^{-\alpha}r−α.
For \alpha =0α=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 >1α>1
there is a coarsening behaviour with competing domains of opposite sign
without development of magnetisation. For strong long range,
i.e. 0<\alpha <10<α<1,
we show that the system shows both features, with probability
P_\alpha (N)Pα(N)
of having the latter one, with the different limiting behaviours
\lim _{N\to \infty}P_\alpha (N)=0limN→∞Pα(N)=0
(at fixed \alpha<1α<1)
and \lim _{\alpha \to 1}P_\alpha (N)=1limα→1Pα(N)=1
(at fixed finite NN).
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^\alphaτα(N)∼Nα.
Correlation bound for a one-dimensional continuous long-range Ising model
