Nonconserved critical dynamics of the two-dimensional Ising model as a surface kinetic roughening process

TL;DR

This work treats the nonconserved critical dynamics of the 2D Ising universality class as a surface kinetic roughening problem, analyzed through both lattice Glauber dynamics and the TDGL equation, and complemented by an integral GL model. It demonstrates that the ordering of initial conditions at criticality governs the dynamic scaling: ordered-quench dynamics follow Family-Vicsek scaling, while disordered-quench dynamics exhibit an initial overgrowth with intrinsic anomalous roughening before relaxing to equilibrium. The study also shows that time-independent, universal fluctuation PDFs emerge when fluctuations are rescaled by the time-dependent roughness, paralleling themes from KPZ-like systems, and reveals a symmetry in fluctuation statistics under space integration in the integral GL framework. Together, these results quantify the kinetic roughening exponents ($\alpha$, $z$, $\eta$) and show how linear instabilities at criticality are stabilized by nonlinear terms, with the integral GL model reinforcing the same qualitative picture and offering a positive roughness exponent. The findings advance the understanding of non-equilibrium critical dynamics and offer robust tools (scaling, PDFs, and integration mappings) for identifying universality beyond traditional exponent analysis.

Abstract

We have revisited the non-conserved (or model A) critical dynamics of the two-dimensional Ising model through numerical simulations of its lattice and continuum formulations --Glauber dynamics and the timedependent Ginzburg-Landau (TDGL) equation, respectively--, to analyze them with current tools from surface kinetic roughening. Our study examines two critical quenches, one from an ordered and a different one from a disordered initial state, for both of which we assess the dynamic scaling ansatz, the critical exponent values, and the fluctuation field statistics that occur. Notably, the dynamic scaling ansatz followed by the system strongly depends on the initial condition: a critical quench from the ordered phase follows Family-Vicsek (FV) scaling, while a critical quench from the disordered phase shows an initial overgrowth regime with intrinsic anomalous roughening, followed by relaxation to equilibrium. This behavior is explained in terms of the dynamical instability of the stochastic Ginzburg-Landau equation at the critical temperature, whereby the linearly unstable term is eventually stabilized by nonlinear interactions. For both quenches we have determined the occurrence of probability distribution functions for the field fluctuations, which are time-independent along the non-equilibrium dynamics when suitably normalized by the time-dependent fluctuation amplitude. Additionally, we have developed a related interface model for a field which scales as the space integral of the TDGL field (integral GL model). Numerical simulations of this model reveal either FV or faceted anomalous roughening, depending on the critical quenched performed, as well as an emergent symmetry in the fluctuation statistics for a critical quench from the ordered phase.

Figures (32)

• Figure 1: $\mathcal{A}$ denotes the mapping whereby the local magnetization $\phi(\mathbf{r},t)$ of the 2D Ising model (left panel) at the critical temperature is interpreted as the height field of a kinetically rough surface (middle panel). This mapping is studied in Secs. \ref{['sec:T=0']} and \ref{['sec:T=infty']}. Then, $\mathcal{B}$ denotes the mapping from such a surface to the integral GL model, a different but related rough surface $h(\mathbf{r},t)$, imaged on the right panel and addressed in Sec. \ref{['sec:im']}.
• Figure 2: (a) Surface roughness as a function of time from simulations of the TDGL model after a critical quench from $T=0$. The solid line corresponds to the fit $W_{\rm fit}=0.76-0.37\; t^{-0.14}$. (b) Same as panel (a) for Glauber dynamics.
• Figure 3: (a) Structure factor from simulations of the TDGL model for different times after a critical quench from $T=0$. Dashed line indicates asymptotic behavior $S(q) \sim q^{-1.80}$. (b) Same as panel (a) for Glauber dynamics. Dashed line indicates asymptotic behavior $S(q) \sim q^{-1.75}$. For both panels, time increases from blue ($t=0$) to red ($t=20000$) being log-spaced for different curves.
• Figure 4: (a) Collapse of the data in Fig. \ref{['fig:Sq_t0']}(a) for the TDGL model following the FV dynamic scaling ansatz, Eqs. \ref{['ec:SFV']}-\ref{['ec:SFVf']}. The dashed line corresponds to $f_S \sim u^{1.80}$ and the solid line represents $f_S = \text{cnst}$. (b) Same as in panel (a) for the Glauber dynamics data in Fig. \ref{['fig:Sq_t0']}(b). The dashed line corresponds to $f_S \sim u^{1.75}$ and the solid line represents $f_S = \text{cnst}$. In both panels, time increases from blue to red (being linearly-spaced) in the growth regime $(t<10^4 )$, and the exponent values employed for collapse are $\alpha=-1/8$ and $z=2.19$.
• Figure 5: (a) Time power spectrum as a function of frequency from simulations of the TDGL model after a critical quench from $T=0$ (blue solid line). The dashed orange line corresponds to $\Omega(\omega) \sim \omega^{-1.805}$. (b) Same as panel (a) for Glauber dynamics (blue solid line). The dashed orange line corresponds to $\Omega(\omega) \sim \omega^{-1.80}$.