Short-time dynamics in phase-ordering kinetics

Abstract

Short-time dynamics in the $2D$ Blume-Capel model, with a non-conserved order-parameter and short-ranged interactions, is analysed. For non-equilibrium dynamics, both at a critical point in the $2D$ Ising universality class and at the tricritical point, we reproduce the values $Θ=0.190({5})$ and $Θ=-0.542({5})$, respectively, of the critical initial slip exponent. These agree with more early estimates and with the Janssen-Schaub-Schmittmann scaling relation. In phase-ordering kinetics, after a quench into the ordered phase, we establish the validity of short-time dynamics. In the $2D$ Ising universality class, we find $Θ=0.39({1})$ in agreement with the scaling relation $λ=d-2Θ$.

Figures (11)

• Figure 1: Magnetisation in the $2D$ Blume-Capel model, for a periodic square lattice $L\times L$ with $L=80$ and the initial magnetisation $m_0=0.1$. Left panel: Initial growth $M(t)\sim t^{\Theta}$ at the critical point $P_c$ (at $T=T_c$, $\Delta_c=0$) and the cross-over towards the equilibrium-controlled decay. The initial slip exponent is $\Theta\approx 0.19$. Right panel: Initial growth below the critical point, for $T=[0.4T_c, 0.6T_c, 0.8T_c]$ from top to bottom, and the cross-over towards saturation. The initial slip exponent is $\Theta\approx 0.39$.
• Figure 2: Phase diagram of the $2D$ Blume-Capel model. The prediction (\ref{['eq:BC-lignecrit']}) is the full line for the second-order transitions and the dashed line for the first-order transitions, separated by the tricritical point $P_{\rm t}$ at $(T_{\rm t}=0.608, \Delta_{\rm t}=1.966)$ (violet diamond). The Ising critical point $P_c$ is at $(T=T_c=1.6929, \Delta=0)$ (red dot). The three blue dots at $\Delta=0$ give the locations for the low-temperature measurements. The small symbols on the critical line indicate numerical estimates, obtained from Wang-Landau Monte Carlo simulation (red square $\blacksquare$Silva, open magenta circle $\circ$Kwak15), microcanonical annealing (blue circle $\bullet$Mozo24), high-temperature series (green star $\ast$Bute18) and hybrid Monte Carlo (orange square $\blacksquare$Zier17).
• Figure 3: Quench onto the point $P_c$, for $L=80$ (red upper curve) and $L=160$ (blue lower curve) and initial magnetisation $m_0$. Left panel: Magnetisation $M(t)$ for $m_0=0.1$. Fit range: $t=2-600$. Centre panel: Squared magnetisation $M^{(2)}(t)$ for $m_0=0.1$. Fit range: $t=2-600$. Right panel: Squared magnetisation $M^{(2)}(t)$ for $m_0=0$. Fit range: $t=30-600$.
• Figure 4: Quench onto the point $P_c$, for $L=80$ (red lower curve) and $L=160$ (blue upper curve) and initial magnetisation $m_0$. Left panel: Global correlator $C(t)$ for $m_0=0.1$. Fit range: $t=2-600$. Centre panel: Global correlator $C(t)$ for $m_0=0$. Fit range: $t=2-600$. Right panel: auto-correlator $A(t)$ for $m_0=0$. Fit range: $t=20-600$.
• Figure 5: Evolution after a quench onto the tricritical point $P_{\rm t}$, with $L=80$ (blue lower curve) and $L=160$ (green upper curve) and $m_0$. Left panel: Magnetisation $M(t)$ for $m_0=0.1$. Fit range: $t=250-2000$. Centre panel: Global correlator $C(t)$, for $m_0=0.1$. Fit range: $t=250-2000$. Right panel: Local auto-correlator $A(t)$ for $m_0=0$. Fit range: $t=100-800$.