We study the persistence properties of the
T=0 coarsening dynamics of one-dimensional q-state Potts model using a modified mean-field approximation (MMFA). In this approximation, the spatial correlations between the interfaces separating spins with different Potts states is ignored, but the correct time dependence of the mean density
P(t) of persistent spins is imposed. For this model, it is known that
P(t) follows a power-law decay with time, $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}(q)},$ where $\ensuremath{\theta}(q)$ is the q-dependent persistence exponent. We study the spatial structure of the persistent region within the MMFA. We show that the persistent site pair correlation function
P2(r,t) has the scaling form
P2(r,t)=P(t)2f(r/t1/2) for all values of the persistence exponent $\ensuremath{\theta}(q).$ The scaling function has the limiting behavior $f(x)\ensuremath{\sim}{x}^{\ensuremath{-}2\ensuremath{\theta}}$ $(x\ensuremath{\ll}1)$ and $f(x)\ensuremath{\rightarrow}1$ $(x\ensuremath{\gg}1).$ We then show within the independent interval approximation (IIA) that the distribution
n(k,t) of separation k between two consecutive persistent spins at time t has the asymptotic scaling form ${n(k,t)=t}^{\ensuremath{-}2\ensuremath{\varphi}}{g(t,k/t}^{\ensuremath{\varphi}}),$ where the dynamical exponent has the form $\ensuremath{\varphi}=\mathrm{max}(1/2,\ensuremath{\theta}).$ The behavior of the scaling function for large and small values of the arguments is found analytically. We find that for small separations $k\ensuremath{\ll}{t}^{\ensuremath{\varphi}},n(k,t)\ensuremath{\sim}{P(t)k}^{\ensuremath{-}\ensuremath{\tau}},$ where $\ensuremath{\tau}=\mathrm{max}[2(1\ensuremath{-}\ensuremath{\theta}),2\ensuremath{\theta}],$ while for large separations $k\ensuremath{\gg}{t}^{\ensuremath{\varphi}},$
g(t,x) decays exponentially with x. The unusual dynamical scaling form and the behavior of the scaling function is supported by numerical simulations.
For the Glauber dynamics of the one-dimensional q-state Potts model at zero temperature, the fraction of spins which have never flipped up to time t decreases as t - theta (q ) where the exponent theta (q) varies continuously with q. The calculation of the q dependence of this exponent can be reduced to the study of a simple reaction-diffusion model. Using finite-size scaling, this allows one to obtain accurate estimates of B(q) for all values of q. One can also show that for 1=1+ epsilon , the expansion of theta (y) is given by theta (1)=(2/ pi ) epsilon +O( epsilon 2 ).
The distribution n(k,t) of the interval sizes k between clusters of persistent sites in the dynamical evolution of the one-dimensional q-state Potts model is studied using a combination of numerical simulations, scaling arguments, and exact analysis. It is shown to have the scaling form n(k,t)=t(-2z)f(k/t(z)), with z=max(1/2, straight theta), where straight theta(q) is the persistence exponent which describes the fraction P(t) approximately t(-straight theta) of sites which have not changed their state up to time t. When straight theta>1/2, the scaling length t(straight theta) for the interval-size distribution is larger than the coarsening length scale t(1/2) that characterizes spatial correlations of the Potts variables.
Critical scaling and universality in short-time dynamics for spin models on a twodimensional triangular lattice are investigated by using Monte Carlo simulation. Emphasis is placed on the dynamic evolution from fully ordered initial states to show that universal scaling exists already in the short-time regime in form of power-law behavior of the magnetization and Binder cumulant. The results measured for the dynamic and static critical exponents, θ, z, β and ν, confirm explicitly that the Potts models on the triangular lattice and square lattice belong to the same universality class. Our critical scaling analysis strongly suggests that the simulation for the dynamic relaxation can be used to determine numerically the universality.
We study the nonequilibrium dynamics of the q-state Potts model following a quench from the high-temperature disordered phase to zero temperature. The time-dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale L(t)\ensuremath{\sim}
t1/2. In particular, the autocorrelation function decays as [L(t)${]}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}(\mathit{q})}$. We illustrate these properties by solving exactly the kinetic Potts model in d=1. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general q and d. We establish a correspondence between the two-point correlations of the q-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization (2/q-1). The dynamics of this Ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter. For general q and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent \ensuremath{\lambda}(q). These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.
The critical coarsening dynamics of the spin S =1/2, 3/2 antiferromagnetic Ising model on a triangular lattice is studied by the dynamic Monte Carlo simulation using a heat bath algorithm. The triangular antiferromagnetic Ising (TAI) model possesses an intrinsic geometrical frustration and a large degeneracy of ground state which may affect the equilibrium and non-equilibrium critical behaviors. The S =1/2 TAI has no phase transition at a finite temperature, but it was suggested that the S =3/2 TAI has the Kosterlitz–Thouless (KT)-type phase transition at a finite temperature. We employ a finite size scaling approach for the correlation function from the short-time dynamics of the S =1/2, 3/2 TAI to calculate the values of the static critical exponent η and the dynamic exponent z. For the S =1/2 TAI, our dynamic scaling analysis provides η =0.498±0.006 and z =2.278±0.020 at T =0, agreeing with the previous results. For the S =3/2 TAI, after identifying a KT-transition temperature TKT =0.51±0.01, we find the temperature-dependent η ranging from 0.301±0.008 at T =0.51 to 0.224±0.016 at T =0 along the KT-line whereas the value of z =2.20±0.06 is constant for T≤TKT. It is shown that the spin S =3/2 TAI model and the two-dimensional XY model, sharing the KT-type phase transition, exhibit similar static critical and coarsening dynamics behavior. For both the S =1/2, 3/2 TAI, the value of z (η) is compatible with (larger than) that of the Ising model at Tc and the XY model for T≤TKT in two-dimension. Our results imply that although the quasi-long-range order disappears with ηXY =0 in the two-dimensional XY model at T =0, the S =3/2 TAI still maintains it with η TAI =0.224 due to the effect of a frustration and a high degeneracy of ground state.