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arXiv:1608.03399 · Physical review. E

Minority-spin dynamics in the nonhomogeneous Ising model: Diverging time scales and exponents

Pratik Mullick, Parongama Sen

Published May 6, 2016• 4 citations
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Abstract

We investigate the dynamical behavior of the Ising model under a zero-temperature quench with the initial fraction of up spins $0\ensuremath{\le}x\ensuremath{\le}1$. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite-size scaling is valid here. In two dimensions, however, the persistence probabilities are no longer algebraic; in particular for $x\ensuremath{\le}0.5$, persistence for the up (minority) spins shows the behavior ${P}_{\text{min}}(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}exp[\ensuremath{-}{(t/\ensuremath{\tau})}^{\ensuremath{\delta}}]$ with time $t$, while for the down (majority) spins, ${P}_{\text{maj}}(t)$ approaches a finite value. We find that the timescale $\ensuremath{\tau}$ diverges as ${({x}_{c}\ensuremath{-}x)}^{\ensuremath{-}\ensuremath{\lambda}}$, where ${x}_{c}=0.5$ and $\ensuremath{\lambda}\ensuremath{\simeq}2.31$. The exponent $\ensuremath{\gamma}$ varies as ${\ensuremath{\theta}}_{2d}+{c}_{0}{({x}_{c}\ensuremath{-}x)}^{\ensuremath{\beta}}$ where ${\ensuremath{\theta}}_{2d}\ensuremath{\simeq}0.215$ is very close to the persistence exponent in two dimensions; $\ensuremath{\beta}\ensuremath{\simeq}1$. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form $E(x)=f\left[(\frac{x\ensuremath{-}{x}_{c}}{{x}_{c}}){L}^{1/\ensuremath{\nu}}\right]$, with $\ensuremath{\nu}\ensuremath{\approx}1.47$. This result suggests that $\ensuremath{\tau}\ensuremath{\sim}{L}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}=\frac{\ensuremath{\lambda}}{\ensuremath{\nu}}=1.57\ifmmode\pm\else\textpm\fi{}0.11$ is an exponent not explored earlier.

DOI
10.1103/physreve.93.052113

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We investigate the dynamical behavior of the Ising model under a zero-temperature quench with the initial fraction of up spins 0\ensuremath≤x\ensuremath≤10\ensuremath{\le}x\ensuremath{\le}10\ensuremath≤x\ensuremath≤1. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite-size scaling is valid here. In two dimensions, however, the persistence probabilities are no longer algebraic; in particular for x\ensuremath≤0.5x\ensuremath{\le}0.5x\ensuremath≤0.5, persistence for the up (minority) spins shows the behavior Pmin(t)\ensuremath∼t\ensuremath−\ensuremathγexp[\ensuremath−(t/\ensuremathτ)\ensuremathδ]{P}_{\text{min}}(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}exp[\ensuremath{-}{(t/\ensuremath{\tau})}^{\ensuremath{\delta}}]Pmin​(t)\ensuremath∼t\ensuremath−\ensuremathγexp[\ensuremath−(t/\ensuremathτ)\ensuremathδ] with time ttt, while for the down (majority) spins, Pmaj(t){P}_{\text{maj}}(t)Pmaj​(t) approaches a finite value. We find that the timescale \ensuremathτ\ensuremath{\tau}\ensuremathτ diverges as (xc\ensuremath−x)\ensuremath−\ensuremathλ{({x}_{c}\ensuremath{-}x)}^{\ensuremath{-}\ensuremath{\lambda}}(xc​\ensuremath−x)\ensuremath−\ensuremathλ, where xc=0.5{x}_{c}=0.5xc​=0.5 and \ensuremathλ\ensuremath≃2.31\ensuremath{\lambda}\ensuremath{\simeq}2.31\ensuremathλ\ensuremath≃2.31. The exponent \ensuremathγ\ensuremath{\gamma}\ensuremathγ varies as \ensuremathθ2d+c0(xc\ensuremath−x)\ensuremathβ{\ensuremath{\theta}}_{2d}+{c}_{0}{({x}_{c}\ensuremath{-}x)}^{\ensuremath{\beta}}\ensuremathθ2d​+c0​(xc​\ensuremath−x)\ensuremathβ where \ensuremathθ2d\ensuremath≃0.215{\ensuremath{\theta}}_{2d}\ensuremath{\simeq}0.215\ensuremathθ2d​\ensuremath≃0.215 is very close to the persistence exponent in two dimensions; \ensuremathβ\ensuremath≃1\ensuremath{\beta}\ensuremath{\simeq}1\ensuremathβ\ensuremath≃1. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form E(x)=f[(x\ensuremath−xcxc)L1/\ensuremathν]E(x)=f\left[(\frac{x\ensuremath{-}{x}_{c}}{{x}_{c}}){L}^{1/\ensuremath{\nu}}\right]E(x)=f[(xc​x\ensuremath−xc​​)L1/\ensuremathν], with \ensuremathν\ensuremath≈1.47\ensuremath{\nu}\ensuremath{\approx}1.47\ensuremathν\ensuremath≈1.47. This result suggests that \ensuremathτ\ensuremath∼Lz\ifmmode~\else˜\fi\ensuremath{\tau}\ensuremath{\sim}{L}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}}\ensuremathτ\ensuremath∼Lz\ifmmode~\else˜\fi, where z\ifmmode~\else˜\fi=\ensuremathλ\ensuremathν=1.57\ifmmode±\else\textpm\fi0.11\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}=\frac{\ensuremath{\lambda}}{\ensuremath{\nu}}=1.57\ifmmode\pm\else\textpm\fi{}0.11z\ifmmode~\else˜\fi=\ensuremathν\ensuremathλ​=1.57\ifmmode±\else\textpm\fi0.11 is an exponent not explored earlier.
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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.
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