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arXiv:cond-mat/9707287 · Journal of Physics A Mathematical and General

Spin block persistence at finite temperature

Stéphane Cueille, Clément Sire

Published Dec 7, 1997• 37 citations
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Abstract

We explore a new definition of the persistence exponent, measuring the probability that a spin never flips after a quench of an Ising-like model at a temperature , while the usual definition only makes sense at T = 0. This probability is now defined for spin blocks, and a general scaling for it, involving time and block linear size is introduced and illustrated by extensive simulations.

DOI
10.1088/0305-4470/30/23/001

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arXiv:1509.07590•Physical review. E•Mar 22, 2016•14 citations
We explore a new definition of the persistence exponent, measuring the probability that a spin never flips after a quench of an Ising-like model at a temperature , while the usual definition only makes sense at T = 0. This probability is now defined for spin blocks, and a general scaling for it, involving time and block linear size is introduced and illustrated by extensive simulations.
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 ttt, while for the down (majority) spins, Pmaj(t){P}_{\text{maj}}(t)Pmaj​(t) approaches a finite value. We find that the timescale $\ensuremath{\tau}$ diverges as ${({x}_{c}\ensuremath{-}x)}^{\ensuremath{-}\ensuremath{\lambda}}$, where xc=0.5{x}_{c}=0.5xc​=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.
We consider a periodic Ising chain with nearest-neighbor and rth neighbor interaction and quench it from infinite temperature to zero temperature. The persistence probability P(t) , measured as the probability that a spin remains unflipped up to time t , is studied by computer simulation for suitable values of r . We observe that as time progresses, P(t) first decays as t(-0.22) (the first regime), then the P(t)-t curve has a small slope (in log-log scale) for some time (the second regime) and at last it decays nearly as t(-3/8) (the third regime). We argue that in the first regime, the persistence behavior is the usual one for a two-dimensional system, in the second regime it is like that of a noninteracting ("zero-dimensional") system, and in the third regime the persistence behavior is like that of a one-dimensional Ising model. We also provide explanations for such behavior.
We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising model, employing the Glauber spin-flip mechanism, in space dimensions d=2d=2d=2 and 333. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures (TfT_fTf​) below the critical one (TcT_{c}Tc​), from different initial temperatures Ti≥TcT_{i} \geq T_{c}Ti​≥Tc​. In long time limit, for Ti>TcT_{i} > T_{c}Ti​>Tc​, the persistence probability exhibits power-law decay with exponents θ≃0.22θ\simeq 0.22θ≃0.22 and ≃0.18\simeq 0.18≃0.18 in d=2d=2d=2 and 333, respectively. For finite TiT_iTi​, the early time behavior is a different power-law whose life-time diverges and exponent decreases as Ti→TcT_{i} \rightarrow T_{c}Ti​→Tc​. The crossover length between the two steps diverges as the equilibrium correlation length. Ti=TcT_i=T_cTi​=Tc​ is expected to provide a {\it{new universality class}} for which we obtain θ≃0.035θ\simeq 0.035θ≃0.035 in d=2d=2d=2 and ≃0.10\simeq 0.10≃0.10 in d=3d=3d=3. The time dependence of the average domain size ℓ\ellℓ, however, is observed to be rather insensitive to the choice of TiT_iTi​.
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.