Domain statistics in the relaxation of the one-dimensional Ising model with strong long-range interactions

After a zero temperature quench, we study the kinetics of the one-dimensional Ising model with long-range interactions between spins at distance $r$ decaying as $r^{-\alpha }$, with $\alpha \le 1$. As shown in our recent study [SciPost Phys 10, 109 (2021)] that only a fraction of the non-equilibrium trajectories is characterized by the presence of coarsening domains while in the remaining ones the system is quickly driven towards a magnetised state. Restricting to realisations displaying coarsening we compute numerically the probability distribution of the size of the domains and find that it exhibits a scaling behaviour with an unusual $\alpha$-dependent power-law decay. This peculiar behaviour is also related to the divergence of the average size of domains with system size at finite times. Such a scenario differs from the one observed when $\alpha >1$, where the distribution decays exponentially. Finally, based on numerical results and on analytical calculations we argue that the average domain size grows asymptotically linearly in time.