Out-of-equilibrium spinodal-like scaling behaviors across the magnetic first-order transitions of 2D and 3D Ising systems

TL;DR

This study analyzes non-equilibrium scaling when 2D, 3D Ising systems are slowly driven across magnetic first-order transitions using Kibble-Zurek protocols with $h(t)=t/t_s$ at fixed $T<T_c$. It distinguishes finite-size scaling (OFSS) and thermodynamic-limit (TL) regimes, deriving scaling variables and showing spinodal-like behavior in the TL with dimension-dependent logarithmic scaling: $h_*$ decays as $1/(\ln t_s)^{\kappa}$ with $\kappa=2$ (2D) and $\kappa=1$ (3D); 2D TL exhibits a singular scaling near a crossing point with $\sigma_*\approx 3.56$ and $\theta\approx 0.12$, while 3D TL follows a smoother scaling with $\sigma=t(\ln t)/t_s$. The results are supported by Metropolis and heat-bath dynamics, finite-size data collapse, and a coarse-grained two-state model that captures the OFSS dynamics, highlighting distinct mechanisms across dimensions and offering broader implications for classical first-order transitions.

Abstract

We study the out-of-equilibrium scaling behavior of two-dimensional and three-dimensional Ising systems, when they are slowly driven across their {\em magnetic} first-order transitions at low temperature $T<T_c$, where $T_c$ is the temperature of their continuous transition. We consider Kibble-Zurek (KZ) protocols in which a spatially homogenous magnetic field $h$ varies as $h(t)=t/t_s$ with a time scale $t_s$. The KZ dynamics starts from negatively-magnetized configurations equilibrated at $h_i<0$ and stops at a positive value of $h$ where the configurations acquire a positive average magnetization. We consider the Metropolis and the heat-bath dynamics, which are two specific examples of a purely relaxational dynamics. We focus on two different dynamic regimes. We consider the out-equilibrium finite-size scaling (OFSS) limit in which the system size $L$ and the time scale $t_s$ diverge simultaneously, keeping an appropriate combination fixed. Then, we analyze the KZ dynamics in the thermodynamic limit (TL), obtained by taking first the $L\to\infty$ limit at fixed $t$ and $t_s$, and then considering the scaling behavior in the large-$t_s$ limit. Our numerical results provide evidence of OFSS, as predicted by general scaling arguments. The results in the TL show the emergence of spinodal-like behaviors: The passage from the negatively-magnetized phase to the positively-magnetized one occurs at positive values $h_*>0$ of the magnetic field, which decrease as $h_* \sim 1/(\ln t_s)^κ$, with $κ= 2$ and $κ=1$ in two and three dimensions, respectively, for $t_s\to\infty$. We identify $σ\equiv t (\ln t)^κ/t_s$ as the relevant scaling variable associated with the KZ dynamics in the TL.

Figures (16)

• Figure 1: Snapshots of a 2D Ising system of size $L=1024$, for different values of $t$. Results for a heat-bath KZ evolution with $t_s = 10000$. A red dot corresponds to a positive spin. Time increases moving from left to right along a row. The leftmost top panel shows the system in the metastable state ($m \approx -m_0$). The second panel on the left (top row) shows the system just before it changes phase (here $m\approx -0.70$).
• Figure 2: Average time evolution of the magnetization as a function of $\Phi =h(t) L^2$. 2D Ising results at $\beta=1.2\beta_c$ for $\Upsilon=0.01$ (bottom) and 0.001 (top). Errors on $m(t)$ are at most 0.0002.
• Figure 3: Results for the 2D Ising model at $\beta=1.2\beta_c$. Bottom: Plot of the magnetization for one specific heat-bath evolution for $L=14$ and $t_s = 4300740$ ($\Upsilon = 0.01$), as a function of $\Phi=h(t) L^2$. Top: Spin configuration when the system is changing phase $(M=0)$. Squares label the lattice sites where $s_{\bm x} = 1$. We remind the reader that we are using periodic boundary conditions.
• Figure 4: Scaling behavior of the average bond-energy density at $\beta=1.2\beta_c$, for $\Upsilon = 0.01$, see Eq. (\ref{['DeltaE-scaling']}). Results for the 2D Ising model. We also report the prediction (blue solid curve) $c \,\Phi [1 - M(h)]$, obtained using the $L=14$ data. We set $c = -0.40$ (obtained by fitting the data).
• Figure 5: Average time evolution of the magnetization as a function of $\Phi =h(t) L^3$. Results for the 3D Ising model (heat-bath dynamics) at $\beta=0.242$ for $\Upsilon=0.0002$.