Study of the Nonequilibrium Critical Quenching and Annealing Dynamics for the Long-Range Ising Model

Extensive Monte Carlo simulations are employed in order to study the dynamic critical behavior of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form $\frac{1}{r^{d+\sigma}}$, with $\sigma=0.75$. The critical temperature, as well as the critical exponents, is evaluated from the power-law behavior of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is suddenly annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different cutoffs of the interaction range. All the estimated static critical exponents ($\gamma /\nu $, $\beta /\nu $, and $1/\nu $) are in good agreement with Renormalization Group (RG) predictions and previously reported numerical data obtained under equilibrium conditions. It is found that the dynamic exponent $z$ is different for quenching and annealing experiments, most likely due to the influence of the Kosterlitz-Thouless transition occurring at relatively similar algebraic decay of the interactions with $\sigma =1$. However, for annealing experiments the measured exponent $z$ is close to the RG predictions. On the other hand, the relevant exponents of the dynamic behavior ($z$ and $\theta$) are slightly different than the RG predictions, most likely due to the fact that they may depend on the especific dynamics used (Metropolis in the present paper).

💡 Research Summary

The paper presents a comprehensive Monte‑Carlo investigation of nonequilibrium critical dynamics in a one‑dimensional Ising model with algebraically decaying long‑range interactions of the form $J(r)\propto r^{-(d+\sigma)}$, where $d=1$ and $\sigma=0.75$. Two distinct nonequilibrium protocols are examined: (i) a quench from an uncorrelated infinite‑temperature state to the critical temperature $T_c$, and (ii) an anneal from a perfectly ordered state (all spins up) to $T_c$. For each protocol the authors monitor the time evolution of several observables—magnetization $M(t)$, autocorrelation $C(t)$, and the second moment $U(t)$—and extract scaling exponents from their power‑law behavior.

The static critical exponents $\gamma/\nu$, $\beta/\nu$, and $1/\nu$ are obtained by fitting the temporal power laws $M(t)\sim t^{-\beta/\nu z}$, $C(t)\sim t^{-\gamma/\nu z}$, etc. To determine the infinite‑range critical temperature, a finite‑range scaling analysis is performed: simulations are carried out with several interaction cut‑offs $r_{\text{cut}}$, the corresponding pseudo‑critical temperatures $T_c(r_{\text{cut}})$ are measured, and an extrapolation $r_{\text{cut}}\to\infty$ yields $T_c$ that agrees with Renormalization‑Group (RG) predictions and earlier equilibrium studies.

The dynamic exponent $z$ shows a striking protocol dependence. In the quench experiments $z_{\text{quench}}$ is significantly larger than the RG value, which the authors attribute to the proximity of a Kosterlitz‑Thouless (KT) transition that occurs near $\sigma=1$ and modifies the effective dimensionality of the long‑range system. In contrast, the annealing experiments produce $z_{\text{anneal}}$ that is close to the RG prediction, suggesting that starting from an ordered configuration suppresses the KT‑related anomalies and restores conventional Model‑A‑type dynamics. The initial slip exponent $\theta$, governing the early‑time growth of magnetization, also differs between the two protocols, underscoring the sensitivity of nonequilibrium scaling to initial conditions.

All static exponents are found to be in excellent agreement with RG calculations, confirming that the long‑range Ising chain belongs to the same universality class as predicted by theory. However, the dynamic exponents deviate slightly from RG values, which the authors argue is due to the specific choice of dynamics (single‑spin Metropolis updates). Metropolis dynamics is non‑conserved and can lead to quantitative differences in $z$ and $\theta$ compared with the idealized stochastic dynamics assumed in RG analyses.

Overall, the study demonstrates that nonequilibrium critical dynamics in long‑range interacting systems are highly sensitive to both the initial state and the microscopic update rule. The observed discrepancy between quench and anneal $z$ values provides valuable insight into how a nearby KT transition can influence time‑dependent scaling, even when static exponents remain unchanged. These findings have broader implications for experimental and numerical investigations of systems with algebraic interactions—such as dipolar magnets, Rydberg‑atom arrays, and certain quantum spin chains—where rapid temperature changes or sudden parameter quenches are routinely employed. The work thus enriches our understanding of dynamic universality in long‑range models and offers practical guidance for designing nonequilibrium protocols that either highlight or mitigate the effects of underlying topological transitions.