Coarsening kinetics in spin systems with long-range interactions: from voter to Ising

TL;DR

This work analyzes coarsening kinetics in one-dimensional spin systems with long-range interactions, contrasting the LR Ising model with the long-range voter model and introducing the p-voter model to interpolate between them. Using analytical arguments based on domain-wall dynamics, scaling forms, and numerical results, it identifies distinct growth laws for the coarsening length $L(t)$ across three regimes of the decay exponent $\alpha$ and shows that the two foundational models fall into different universality classes in 1D LR settings. The p-voter model, which reduces to the voter model at $p=1,2$ and to Ising-like behavior for $p\ge3$, captures Ising-like coarsening at finite temperature and converges to zero-temperature IM behavior as $p\to\infty$, providing a tractable interpolation between the two paradigms. The findings highlight the nontrivial impact of nonlocal interactions on ordering kinetics and establish the pVM as a useful proxy for exploring LR-coarsening phenomena across related models.

Abstract

In this paper, we start reviewing the main features of the one-dimensional Ising model with long-range interactions, where the spin-spin coupling decays as a power law, $J(r) \propto r^{-α}$. We then discuss the key properties of the one-dimensional voter model, in which two agents (spins) at distance $r$ interact with a power-law probability with the same form of $J(r)$. The two models are compared, and the so-called $p$-voter model is presented, which provides a framework to interpolate between them. Specifically, the $p$-voter model reduces to the voter model for $p = 1$ and $p = 2$, while for $p \ge 3$ it falls into the universality class of the Ising model.

Figures (9)

• Figure 1: Simple one dimensional configuration, with periodic boundary conditions, with a single domain of down spins and length $X(t)$ (with $X(0)=L$). See the main text for further details.
• Figure 2: $L(t)$ for a system quenched from $T_i=\infty$ to $T=10^{-3}$ on a double-logarithmic scale. Different symbols and colors correspond to different values of $\alpha$ and to the nn case (see legend). The dashed orange line is the $t^{1/2}$ law and the dashed green one is the ballistic behavior. The color dotted lines (below the data curves) are the power-laws $t^{1/\alpha}$ for each $\alpha$ value CLP_review.
• Figure 3: $L(t)$ for a system quenched from $T_i=\infty$ to different final $T$ for $\alpha=1.5$. The green dashed line indicate the linear, ballistic regime $L(t)\sim t$. The magenta dashed line is the growth $L\sim t^{1/\alpha}=t^{2/3}$ in the slow regime. The system size is $N=8\times 10^8$CLP_review.
• Figure 4: $L(t)$ is plotted against time $t$, for different values of $\alpha$ (see legend). Data are obtained by solving numerically Eq. (\ref{['eqc2']}) with $N=10^3$. Dashed lines (with corresponding color as numerical data) are the analytical results of Eqs. (\ref{['diffgrowth']},\ref{['alphareg']},\ref{['balreg']}). In the inset a comparison is shown, for $\alpha=2.5$ and $\alpha=1.8$, with a larger system with $N=10^4$ (dot-dashed lines). Note that all curves saturate to $N/4$, which corresponds, according to Eq. \ref{['eqL']}, to the fully ordered state CorbCast24.
• Figure 5: Collapse plot, for $\alpha=3.5$, of the correlation function $C(r,t)$ as a function of $x=r/L(t)$ (main plot), or $x=r/\mathcal{L}(t)$ (inset). $L(t)$ is computed using Eq. \ref{['eqL']} and $\mathcal{L}(t)$ as defined in the text. System size is $N=10^4$. The curves are for values of $t$ ranging from $10^2$ to $10^3$ in steps of $10^2$CorbCast24.