Long-term ordering kinetics of the two-dimensional $q$-state Potts model

We studied the nonequilibrium dynamics of the $q$-state Potts model in the square lattice, after a quench to subcritical temperatures. By means of a continuous time Monte Carlo algorithm (nonconserved order parameter dynamics) we analyzed the long term behavior of the energy and relaxation time for a wide range of quench temperatures and system sizes. For $q>4$ we found the existence of different dynamical regimes, according to quench temperature range. At low (but finite) temperatures and very long times the Lifshitz-Allen-Cahn domain growth behavior is interrupted with finite probability when the system gets stuck in highly symmetric nonequilibrium metastable states, which induce activation in the domain growth, in agreement with early predictions of Lifshitz [JETP 42, 1354 (1962)]. Moreover, if the temperature is very low, the system always gets stuck at short times in highly disordered metastable states with finite lifetime, which have been recently identified as glassy states. The finite size scaling properties of the different relaxation times involved, as well as their temperature dependency, are analyzed in detail.

Figure 1

(Color online) Relaxation function ${\varphi }_E\left(t\right)$ for $L=300$ and $q=9$ and temperatures $T=0.5$ (bottom), 0.6, 0.64, 0.66, 0.68, 0.7, 0.715, 0.719, and 0.72 (top).

Figure 2

(Color online) Equilibration time probability distribution $P\left(\tau \right)$ for $L=100$ and $q=9$. (a) Temperatures ranging from $T=0.4$ (top) to 0.719 (bottom). (b) Temperatures ranging from $T=0.3$ (left) to 0.09 (right).

Figure 3

(Color online) Characteristic relaxation times ${\tau }_1$ and ${\tau }_2$ vs $1∕T$ for $L=100$ and $q=9$; the same qualitative behaviors are observed for other system sizes. The continuous lines are a guide to the eye.

Figure 4

(Color online) Energy per spin as a function of time and typical spin configurations in one realization of the stochastic noise, when the system gets stuck in a striped configuration ($L=200$, $q=9$, and $T=0.2$). The different colors codify different spin values $s_i=1,\dots ,9$.

Figure 5

(Color online) Equilibration time probability distribution $P\left(\tau \right)$ for $T=0.2$ and $q=9$ and system sizes ranging from $L=15$ (left) to 500 (right).

Figure 6

(Color online) Characteristic relaxation time ${\tau }_1$ vs $L$ for $q=9$ and different temperatures; dotted lines are a guide to the eye.

Figure 7

(Color online) Characteristic relaxation time ${\tau }_2$ vs $L$ for $T=0.2$ and different values of $q$. The dashed lines correspond to linear fittings.

Figure 8

(Color online) Finite size scaling exponent $\omega$ of the characteristic relaxation time ${\tau }_2$ as a function of $T$ for different values of $q$; dotted lines are a guide to the eye.

Figure 9

(Color online) Equilibration time probability distribution $P\left(\tau \right)$ starting from different initial configurations (arbitrary normalization) for $L=300$ and $T=0.2$. The inset images show typical blocked spin configurations at the corresponding times.

Figure 10

Characteristic relaxation time ${\tau }_3$ vs temperature for $q=9$ and $L=100$; the continuous line is a guide to the eye.

Figure 11

(Color online) Basic relaxation mechanism of a Lifshitz state at low temperatures. The circle in (b) marks the creation of a dent in the vertex of (a), by flipping a spin from $2\to 3$ with an energy cost $\Delta E=1$. Circles in (c) and (d) exemplify spins that can flip without energy cost $(\Delta E=0)$ along the three converging walls. The whole process may lead to the hopping of the whole structure (vertex plus walls) and ultimately to the collapse of two vertices.

Figure 12

(Color online) Probability of getting stuck in a blocked state $P_b\left(q\right)$ for $q=9$: (a) as a function of $L$ for different temperatures and (b) as a function of $T$ for different sizes. The dashed line corresponds to a linear fitting of the points for $L=200$, giving an extrapolated value of $0.008\pm 0.01$ at $T=0$.

Figure 13

Probability of getting stuck in a blocked state $P_b\left(q\right)$ for $L=200$ and $T=0.15$.

Figure 14

(Color online) Relaxation function for $q=9$, $L=200$, and temperatures 0.3 (left), 0.2, 0.12, 0.1, 0.09, and 0.07 (right).

Figure 15

(Color online) Relaxation function for $q=9$, $T=0.1$, and different values of $L$. The inset shows a typical configuration of the glassy state associated with the plateau.

Figure 16

(Color online) Equilibration time probability distribution $P\left(\tau \right)$ for $q=9$, $L=200$, and $T=0.2$, using periodic (p.b.c) and open (o.b.c) boundary conditions. The inset images show typical blocked spin configurations observed with o.p.c.

Figure 17

Dynamical regimes in the long term relaxation of the $q$-state Potts model with $q>4$, after a quench from infinite temperatures down to a subcritical temperature $T$.