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

TL;DR

This paper investigates the dynamics of a one-dimensional long-range Ising model after a low-temperature quench, revealing a coexistence of coarsening and mean-field relaxation behaviors depending on the interaction range parameter .

Contribution

It uncovers the simultaneous presence of coarsening and mean-field relaxation in the long-range Ising chain and characterizes their dependence on the interaction decay parameter  and system size.

Findings

For =0, the system quickly reaches a magnetized state.

For >1, coarsening dominates without magnetization.

For 0<<1, both behaviors coexist with probabilities depending on system size and .

Abstract

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