Critical aging and relaxation dynamics in long-range systems

TL;DR

The paper develops a non-perturbative, non-equilibrium functional RG framework to study critical aging and relaxation in long-range $O(N)$ models after a quench to the critical temperature. It computes the dynamical exponent $z$ and aging exponent $\theta$ across the full LR range $\sigma>0$ and for various $d$ and $N$, benchmarking against Monte Carlo data and the large-$N$ limit, and validating the LR-to-SR effective-dimension correspondence for dynamics. A key finding is that long-range interactions generally speed up relaxation (lower $z$) and can enhance the performance-rate exponent $\pi_{th}=\alpha-z\nu$ in critical heat-engine cycles, despite reductions in the static exponent $\alpha$, indicating a net thermodynamic advantage in certain regimes. The results provide a unified scaling picture of non-equilibrium critical aging in LR systems and suggest practical applications in finite-time thermodynamics, while also outlining avenues for methodological improvements and extensions to more complex settings.

Abstract

We study the dynamical scaling of long-range $\mathrm{O}(N)$ models after a sudden quench to the critical temperature, using the functional renormalization group approach. We characterize both short-time aging and long-time relaxation as a function of the symmetry index $N$, the interaction range decay exponent $σ$ and the dimension $d$. Our results substantially improve on perturbative predictions, as demonstrated by benchmarks against Monte Carlo simulations and the large-$N$ limit. Finally, we demonstrate that long-range systems increase the performance of critical heat engines with respect to a local active medium.

Figures (5)

• Figure 1: Dynamical exponents $z$ (upper panel) and $\theta'$ (lower panel) for the long-range Ising model in $d=1$. The dark orange dots are obtained using the fRG scheme described in \ref{['sec:frg']}, with the regulator \ref{['Litim regulator']}. The dotted lines show the weak-coupling expansions of chen2000short, which is only valid in the vicinity of $\sigma = 0.5$. The dots with error bars are MC estimates from Ref. uzelac2008short (black), Ref. tomita2008monte (red), and, as explained in the main text, Refs. grassberger1995damagenightingale2000montehasenbusch2020dynamic via the effective dimension equivalence (blue).
• Figure 2: Performance-rate exponent $\pi_{\rm th} = \alpha-z\nu$ for long-range (LR) and short-range (SR) Ising models. The horizontal red and blue dashed lines correspond to the SR Ising model in dimension $D=2$ and $D=3$, respectively. The brown and yellow lines represent the values of the exponent $\pi_{\rm th}$ as $\sigma$ is varied for the LR Ising model in dimension $d=1$ and $d=2$, respectively. The latter curves are obtained by interpolating fRG data points, shown as crosses. The mean-field value $\pi_{\rm th} = -1$, reached by all long-range models at $\sigma_{\rm mf}=d/2$, is visualized as a horizontal green dash-dotted line.
• Figure 3: Aging exponent $\theta = \theta(\sigma)$ for the long-range $\mathrm{O}(N)$ models in $d=2$ with $N=1,3,10,100$. The colored dots represent data points obtained via the fRG scheme. The large-$N$ limit \ref{['aging exponent largeN']} is denoted by the solid violet line. The dashed lines are $\theta_{\text{SR}}(D(\sigma))$ for $N=1,3,10$ obtained from the short-range model through the effective dimension approach of \ref{['sec:effdim']}. The horizontal dashed line for $N=1$ starting at $\sigma \approx 1.75$ shows the role of the short-range term in the Ising case for large values of $\sigma$. The blue dots are the effective dimension Monte Carlo estimates for $N=1$, obtained from the short-range model via \ref{['mapping MC a']}, \ref{['mapping MC c']} and \ref{['effective sigma']}.
• Figure 4: Dynamical exponent $z = z(\sigma)$ for the long-range $\mathrm{O}(N)$ models in $d=2$ with $N=1,3,10,100$. The colored dots represent data points obtained via the fRG scheme. The large-$N$ limit $z=\sigma$ is denoted by the solid violet line. The dashed lines are $z_{\text{SR}}(D) \, d / D$ -- cf. \ref{['effective dimension z']} and \ref{['effective dimension']} -- for $N=1,3,10$ obtained from the short-range model through the effective dimension approach. The horizontal dashed line for $N=1$ has the same meaning as in \ref{['fig:slip_exponent_all']}. The blue dots are effective dimension Monte Carlo estimates for $N=1$.
• Figure 5: Exponent $\psi$ vs $\sigma$ for the long-range models with $N=1,3,10,100$ in $d=2$. The colored dots represent values obtained from the same fRG data as \ref{['fig:slip_exponent_all']} and \ref{['fig:dynamical_exponent_all']}. The large-$N$ limit $\psi = 2$ is plotted as a violet line.