Dynamical scaling study for the estimation of dynamical exponent $z$ of three-dimensional XY spin glass model

TL;DR

The paper develops and validates a high-precision approach to extract the dynamical exponent $z$ from nonequilibrium relaxation data in spin-glass systems, using a dynamical scaling framework combined with Gaussian process regression to infer the time-dependent correlation length $\xi(t)$. After confirming the method on well-studied 3D FM Ising and 3D $\pm J$ Ising models, the authors apply it to the 3D $\pm J$ XY model, obtaining precise critical temperatures $T_{\mathrm{SG}}$ and $T_{\mathrm{CG}}$, and dynamic exponents $z_{\mathrm{SG}}$ and $z_{\mathrm{CG}}$, along with associated critical exponents and $\eta_{\mathrm{eff}}$. The results provide strong support for spin-chirality decoupling in SG systems and demonstrate improved accuracy over previous NER-based analyses, while highlighting finite-time corrections and the need for robust extrapolation methods. The methodology thus offers a reliable, generalizable tool for dynamical critical phenomena in frustrated magnets and related disordered systems.

Abstract

To analyze the $\pm J$ XY spin-glass in three dimensions, we verified a method aimed at obtaining a high-precision dynamical exponent $z$ from the correlation length in the nonequilibrium relaxation process. The obtained $z$ yielded consistent and highly accurate results in previous studies for relatively well-studied models -- specifically, the three-dimensional (3D) ferromagnetic Ising model and the 3D $\pm J$ Ising model. Building on these previous studies, we used this method and the dynamical scaling method to analyze the 3D $\pm J$ XY model and obtained highly precise critical temperatures and exponents. These findings support the spin chirality decoupling picture, explaining the experimental spin-glass phase transition.

Figures (15)

• Figure 1: Size dependence of the magnetization in the 3D ferromagnetic Ising model. The simulations were performed at the transition temperature $T_{\mathrm{c}} = 4.51152325$Ferrenberg2018. The number of samples is indicated in the legend. It is found that the size dependence can be neglected for $L \ge 301$.
• Figure 2: The dynamical correlation function of the 3D FM Ising model, $f(r,t)$, at the critical temperature $T_{\mathrm{C}}=4.51152325$Ferrenberg2018 for various Monte Carlo steps $t$.
• Figure 3: The scaling plot of the dynamical correlation function $f(r,t)$ for the 3D FM Ising model, based on the dynamical scaling law in Eq. (\ref{['correlation dynamical scaling law']}). The quantities used in the scaling plot are the effective exponent $\eta_{\mathrm{eff}}=0.036(1)$ and the dozens of correlation length $\{\xi(t)\}$ shown in Fig. \ref{['fig:3dising_determinateZ']}. These values were obtained through optimization calculations of Gaussian process regression.
• Figure 4: The correlation length $\{\xi(t)\}$ obtained at the critical point of the 3D FM Ising model. Here, "Inference" in the legend of the graph corresponds to $\{\xi(t)\}$ obtained through optimization calculations of the dynamical scaling law, whereas "Second" represents $\{\xi(t)\}$ obtained using the second moment method. The dotted line indicates the slope corresponding to the estimated dynamic exponent $z$.
• Figure 5: The dynamical SG correlation function of the 3D $\pm J$ Ising model, $f_{\mathrm{SG}}(r,t)$, at the critical temperature $T_{\mathrm{SG}}=1.178$Terasawa2023 for various Monte Carlo steps $t$.