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 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.
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 investigate the universality class of the finite-temperature phase transition of the two-dimensional Ising model with the algebraically decaying ferromagnetic long-range interaction, J_{ij}=|r[over ⃗]_{i}-r[over ⃗]_{j}|^{-(d+σ)}, where d (=2) is the dimension of the system and σ is the decay exponent, by means of the order-N cluster-algorithm Monte Carlo method. In particular, we focus on the upper and lower critical decay exponents, the boundaries between the mean-field-universality, intermediate, and short-range-universality regimes. At the critical decay exponents, it is found that the standard Binder ratio of magnetization at the critical temperature exhibits extremely slow convergence as a function of the system size. We propose more effective physical quantities, namely the combined Binder ratio and the self-combined Binder ratio, both of which cancel the leading finite-size corrections of the conventional Binder ratio. Utilizing these techniques, we clearly demonstrate that in two dimensions, the lower and upper critical decay exponents are σ=1 and 7/4, respectively, contrary to the recent Monte Carlo and renormalization-group studies [M. Picco, arXiv:1207.1018; T. Blanchard et al., Europhys. Lett. 101, 56003 (2013)EULEEJ0295-507510.1209/0295-5075/101/56003].
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=2 and 3. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures (Tf) below the critical one (Tc), from different initial temperatures Ti≥Tc. In long time limit, for Ti>Tc, the persistence probability exhibits power-law decay with exponents θ≃0.22 and ≃0.18 in d=2 and 3, respectively. For finite Ti, the early time behavior is a different power-law whose life-time diverges and exponent decreases as Ti→Tc. The crossover length between the two steps diverges as the equilibrium correlation length. Ti=Tc is expected to provide a {\it{new universality class}} for which we obtain θ≃0.035 in d=2 and ≃0.10 in d=3. The time dependence of the average domain size ℓ, however, is observed to be rather insensitive to the choice of Ti.
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.
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, 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 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 study the time evolution of the two-dimensional kinetic Ising model in finite systems with a non-conserved order parameter, considering nearest-neighbour interactions on the square lattice with periodic and open boundary conditions. Universal data collapse in spin product correlation functions is observed which, when expressed in rescaled units, is valid across the entire time evolution of the system at all length scales, not just within the time regime usually considered in the dynamical scaling hypothesis. Consequently, beyond rapidly decaying finite size effects, the evolution of correlations in small finite systems parallels arbitrarily larger cases, even at large fractions of the size of these finite systems.