Out-of-equilibrium spinodal-like scaling behaviors at the thermal first-order transitions of three-dimensional q-state Potts models

TL;DR

The paper investigates how a three-dimensional $q$-state Potts model behaves when driven across a thermal first-order transition using a linear Kibble-Zurek–like protocol with purely relaxational (heat-bath) dynamics. The authors analyze the time-dependent energy density $E(t)$ in the thermodynamic limit and find that, for large drive times $t_s$, the energy collapses onto a universal function of the scaling variable $σ$, with $σ = t R_{dr}(t)^d / t_s$ and exponent $κ = d/(d-1)$; for $d=3$, $κ = 3/2$, consistent with droplet nucleation as the slow mechanism. A spinodal-like discontinuity at $σ_*$ is observed, and the effective transition shift scales as $δβ_* ∝ (\ln t_s)^{-3/2}$. These results confirm that in 3D Potts FOT dynamics, droplet nucleation sets the longest time scale, in contrast to some Ising cases where different scaling exponents appear, and advance the general understanding of non-equilibrium scaling across first-order transitions in classical and quantum contexts.

Abstract

We study the out-of-equilibrium spinodal-like dynamics of three-dimensional $q$-state Potts systems driven across their thermal first-order transition in the thermodynamic limit, by a relaxational (heat-bath) dynamics. During the evolution, the inverse temperature $β$ increases linearly with time, as $δβ(t)\equiv β(t)- β_{\rm fo} \sim t/t_s$, where $β_{\rm fo}$ is the inverse temperature at the transition point, $t$ is the time and $t_s$ is a time scale. The dynamics starts at $t_i< 0$ from an ensemble of disordered configurations equilibrated at inverse temperature $β(t_i)<β_{\rm fo}$ and ends at positive values of $t$, when the system is ordered (this is analogous to a standard Kibble-Zurek protocol). The time-dependent energy density shows an out-of-equilibrium scaling behavior in the large-$t_s$ limit, in terms of the scaling variable $t(\ln t)^κ/t_s$. The corresponding exponent turns out to be consistent with $κ=3/2$ (with a good accuracy), which is the value obtained by assuming that the initial nucleation of ordered regions provides the relevant mechanism for the passage from one phase to the other. The scaling behavior implies a spinodal-like phenomenon close to the transition point: the passage from the disordered to the ordered phase, composed of large ordered regions of different color, occurs at $δβ(t)=δβ_*>0$, where $δβ_*$ decreases as $1/(\ln t_s)^{3/2}$ in the large-$t_s$ limit.

Figures (6)

• Figure 1: Snapshots of the configurations of a 2D $q=20$ Potts system of size $L=512$, driven across its FOT by a KZ relaxational dynamics (time, and therefore $\beta$, increases from left to right and from top to bottom). The images show the nucleation of different ordered regions, which grow as time increases. Different colors correspond to the $q$ distinct states. These snapshots should give an idea of what happens in 3D Potts models as well.
• Figure 2: The time evolution of the energy density $E(t,t_s,L)$ for $q=6$, versus $\delta(t)=t/t_s$ for various values of $t_s$ and sizes $L$. The comparison of data for different sizes and same $t_s$ (when $t_s$ is not explicitly reported, it is understood that its value is the one comparing in the symbol lines above) shows that the thermodynamic limit is effectively obtained for $L\gtrsim \sqrt{t_s}$ within the very small statistical errors of our simulations.
• Figure 3: The time evolution of the energy density $E(t,t_s,L)$ for $q=10$, versus $\delta(t)=t/t_s$ for several values of $t_s$ and $L$. Comparing data at different $L$ for the same $t_s$ indicates that the thermodynamic limit is reached for $L\gtrsim \sqrt{t_s}$.
• Figure 4: Data for the $q=6$ energy density in the thermodynamic limit versus $\sigma=t (\ln t)^{3/2} /t_s$. The vertical dashed line indicates the estimate $\sigma_*$ of the asymptotic crossing point; the interval between the dotted lines gives the uncertainty.
• Figure 5: The energy density $E(t)$ versus $\hat{\sigma} \equiv (\sigma - \sigma_*)\,t_s^\theta$ for $q=6$, using the optimal values $\sigma_*=0.61$ and $\theta=0.42$. The data appear to approach an asymptotic scaling curve with increasing $t_s$.