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arXiv:cond-mat/0004148 · Journal of Physics A Mathematical and General

Scaling and persistence in the two-dimensional Ising model

S. Jain, H. Flynn

Published Nov 17, 2000• 23 citations
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Abstract

The spatial distribution of persistent spins at zero temperature in the pure two-dimensional Ising model is investigated numerically. A persistence correlation length, ξ( t )~ t Z , is identified such that for length scales r <<ξ( t ) the persistent spins form a fractal with dimension d f ; for length scales r >>ξ( t ) the distribution of persistent spins is homogeneous. The zero-temperature persistence exponent, θ, is found to satisfy the scaling relation θ = Z (2- d f ) with θ = 0.209±0.002 of Jain (Jain S 1999 Phys. Rev. E 59 R2493), Z = 1/2 and d f ~1.58.

DOI
10.1088/0305-4470/33/47/305

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The spatial distribution of persistent spins at zero temperature in the pure two-dimensional Ising model is investigated numerically. A persistence correlation length, ξ( t )~ t Z , is identified such that for length scales r <<ξ( t ) the persistent spins form a fractal with dimension d f ; for length scales r >>ξ( t ) the distribution of persistent spins is homogeneous. The zero-temperature persistence exponent, θ, is found to satisfy the scaling relation θ = Z (2- d f ) with θ = 0.209±0.002 of Jain (Jain S 1999 Phys. Rev. E 59 R2493), Z = 1/2 and d f ~1.58.
The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.
We present very accurate numerical estimates of the time and size dependence of the zero-temperature local persistence in the 2d ferromagnetic Ising model. We show that the effective exponent decays algebraically to an asymptotic value θ that depends upon the initial condition. More precisely, we find that θ takes one universal value 0.199(2) for initial conditions with short-range spatial correlations as in a paramagnetic state, and the value 0.033(1) for initial conditions with the long-range spatial correlations of the critical Ising state. We checked universality by working with a square and a triangular lattice, and by imposing free and periodic boundary conditions. We found that the effective exponent suffers from stronger finite size effects in the former case.
We numerically investigate the heterogeneity in cluster sizes in the two-dimensional Ising model and verify its scaling form recently proposed in the context of percolation problems [Phys. Rev. E 84, 010101(R) (2011)]. The scaling exponents obtained via the finite-size scaling analysis are shown to be consistent with theoretical values of the fractal dimension d(f) and the Fisher exponent τ for the cluster distribution. We also point out that strong finite-size effects exist due to the geometric nature of the cluster-size heterogeneity.
The persistence exponents associated with the T = 0 quenching dynamics of the two-dimensional XY model and a two-dimensional uniaxial spin nematic model have been evaluated using a numerical simulation. The site persistence or the probability that the sign of a local spin component does not change starting from initial time t = 0 up to a certain time t , is found to decay as L ( t ) −θ ( L ( t ) is the linear domain length scale), with θ = 0.305(±0.020) for the two-dimensional XY model and 0.199(±0.009) for the two-dimensional uniaxial spin nematic model. We have also investigated the scaling (at the late time of phase ordering) associated with the correlated persistent sites in both models. The persistence correlation length was found to grow in the same way as L ( t ).
The nonequilibrium dynamics of the strongly diluted random-bond Ising model in two dimensions (2D) is investigated numerically. The persistence probability, P(t), of spins which do not flip by time t is found to decay to a nonzero, dilution-dependent value P(infinity). We find that p(t)=P(t)-P(infinity) decays exponentially to zero at large times. Furthermore, the fraction of spins which never flip is a monotonically increasing function over the range of bond-dilution considered. Our findings, which are consistent with a recent result of Newman and Stein [Phys. Rev. Lett. 82, 3944 (1999)], suggest that persistence in diluted and pure systems falls into different classes. Furthermore, its behavior would also appear to depend crucially on the strength of the dilution present.