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arXiv:1612.05841 · Physical Review E

Time-dependent Monte Carlo simulations of critical and Lifshitz points of the axial-next-nearest-neighbor Ising model

Roberto da Silva, Nelson A. Alves, J. R. Drugowich de Felício

Published Jan 23, 2013• 14 citations
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Abstract

In this work, we study the critical behavior of second-order points, specifically the Lifshitz point (LP) of a three-dimensional Ising model with axial competing interactions [the axial-next-nearest-neighbor Ising (ANNNI) model], using time-dependent Monte Carlo simulations. We use a recently developed technique that helps us localize the critical temperature corresponding to the best power law for magnetization decay over time: ${\ensuremath{\langle}M\ensuremath{\rangle}}_{{m}_{0}=1}\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\beta}/\ensuremath{\nu}z}$, which is expected of simulations starting from initially ordered states. We obtain original results for the dynamic critical exponent $z$, evaluated from the behavior of the ratio ${F}_{2}(t)={\ensuremath{\langle}{M}^{2}\ensuremath{\rangle}}_{{m}_{0}=0}/{\ensuremath{\langle}M\ensuremath{\rangle}}_{{m}_{0}=1}^{2}\ensuremath{\sim}{t}^{3/z}$, along the critical line up to the LP. We explore all the critical exponents of the LP in detail, including the dynamic critical exponent $\ensuremath{\theta}$ that characterizes the initial slip of magnetization and the global persistence exponent ${\ensuremath{\theta}}_{g}$ associated with the probability $P(t)$ that magnetization keeps its signal up to time $t$. Our estimates for the dynamic critical exponents at the Lifshitz point are $z=2.34(2)$ and ${\ensuremath{\theta}}_{g}=0.336(4)$, values that are very different from those of the three-dimensional Ising model (the ANNNI model without the next-nearest-neighbor interactions at the $z$ axis, i.e., ${J}_{2}=0$), i.e., $z\ensuremath{\approx}2.07$ and ${\ensuremath{\theta}}_{g}\ensuremath{\approx}0.38$. We also present estimates for the static critical exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$, obtained from extended time-dependent scaling relations. Our results for static exponents are in good agreement with previous works.

DOI
10.1103/physreve.87.012131

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In this work, we study the critical behavior of second-order points, specifically the Lifshitz point (LP) of a three-dimensional Ising model with axial competing interactions [the axial-next-nearest-neighbor Ising (ANNNI) model], using time-dependent Monte Carlo simulations. We use a recently developed technique that helps us localize the critical temperature corresponding to the best power law for magnetization decay over time: \ensuremath⟨M\ensuremath⟩m0=1\ensuremath∼t\ensuremath−\ensuremathβ/\ensuremathνz{\ensuremath{\langle}M\ensuremath{\rangle}}_{{m}_{0}=1}\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\beta}/\ensuremath{\nu}z}\ensuremath⟨M\ensuremath⟩m0​=1​\ensuremath∼t\ensuremath−\ensuremathβ/\ensuremathνz, which is expected of simulations starting from initially ordered states. We obtain original results for the dynamic critical exponent zzz, evaluated from the behavior of the ratio F2(t)=\ensuremath⟨M2\ensuremath⟩m0=0/\ensuremath⟨M\ensuremath⟩m0=12\ensuremath∼t3/z{F}_{2}(t)={\ensuremath{\langle}{M}^{2}\ensuremath{\rangle}}_{{m}_{0}=0}/{\ensuremath{\langle}M\ensuremath{\rangle}}_{{m}_{0}=1}^{2}\ensuremath{\sim}{t}^{3/z}F2​(t)=\ensuremath⟨M2\ensuremath⟩m0​=0​/\ensuremath⟨M\ensuremath⟩m0​=12​\ensuremath∼t3/z, along the critical line up to the LP. We explore all the critical exponents of the LP in detail, including the dynamic critical exponent \ensuremathθ\ensuremath{\theta}\ensuremathθ that characterizes the initial slip of magnetization and the global persistence exponent \ensuremathθg{\ensuremath{\theta}}_{g}\ensuremathθg​ associated with the probability P(t)P(t)P(t) that magnetization keeps its signal up to time ttt. Our estimates for the dynamic critical exponents at the Lifshitz point are z=2.34(2)z=2.34(2)z=2.34(2) and \ensuremathθg=0.336(4){\ensuremath{\theta}}_{g}=0.336(4)\ensuremathθg​=0.336(4), values that are very different from those of the three-dimensional Ising model (the ANNNI model without the next-nearest-neighbor interactions at the zzz axis, i.e., J2=0{J}_{2}=0J2​=0), i.e., z\ensuremath≈2.07z\ensuremath{\approx}2.07z\ensuremath≈2.07 and \ensuremathθg\ensuremath≈0.38{\ensuremath{\theta}}_{g}\ensuremath{\approx}0.38\ensuremathθg​\ensuremath≈0.38. We also present estimates for the static critical exponents \ensuremathβ\ensuremath{\beta}\ensuremathβ and \ensuremathν\ensuremath{\nu}\ensuremathν, obtained from extended time-dependent scaling relations. Our results for static exponents are in good agreement with previous works.
We have derived a twelfth-order, high-temperature series for the axial next-nearest-neighbor Ising (ANNNI) models on the fcc and the sc lattices, and we present detailed results for one of the fcc models. We have determined the phase boundaries of these models and located the Lifshitz point from the intersection of the phase boundary with the ``disorder'' line, where the maximum in the susceptibility \ensuremath{\chi}(q) begins to occur at nonzero q signaling the onset of modulated correlations. As expected, the Lifshitz point appears to be an inflection point of the phase boundary. The effective susceptibility index shows the predicted universal values for definite regions of the parameter space, e.g., XY-like in the midrange modulated regime, with the expected smooth crossover near the multicritical points. Throughout the modulated regime from the Lifshitz point to the decoupling point, the variation in the critical wave vector shows only small deviations from mean-field predictions. Our best estimates for the Lifshitz-point indices, ${\ensuremath{\gamma}}_{\mathit{l}}$=1.62\ifmmode\pm\else\textpm\fi{}0.12, 4${\ensuremath{\nu}}_{\mathit{l}4}$=1.63\ifmmode\pm\else\textpm\fi{}0.10, and ${\mathrm{\ensuremath{\alpha}}}_{\mathit{l}}$=0.20\ifmmode\pm\else\textpm\fi{}0.15, differ from previous estimates. We are confident that our estimates are more reliable not only because we have longer series for a close-packed lattice but also primarily because we have analyzed our series along the appropriate scaling direction.
In this work the two-dimensional Ising model with nearest- and next-nearest-neighbor interactions is revisited. We obtain the dynamic critical exponents z and θ from short-time Monte Carlo simulations. The dynamic critical exponent z is obtained from the time behavior of the ratio [Formula: see text], whereas the non-universal exponent θ is estimated from the time correlation of the order parameter <M(0)M(t)> ~ tθ, where M(t) is the order parameter at instant t, d is the dimension of the system and <(⋯)> is the average of the quantity (⋯) over different samples. We also obtain the static critical exponents β and ν by investigating the time behavior of the magnetization.
We investigate the dynamics of a two-dimensional biaxial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H=-J_{0}∑_{i,j=1}^{L}[(S_{i,j}S_{i+1,j}+S_{i,j}S_{i,j+1})-κ(S_{i,j}S_{i+2,j}+S_{i,j}S_{i,j+2})]. For κ<1, the system does not reach the equilibrium ground state and keeps evolving in active states forever. For κ≥1, though, the system reaches a final state, but it does not reach the ground state always and freezes to a striped state with a finite probability like a two-dimensional ferromagnetic Ising model and an axial next-nearest-neighbor Ising (ANNNI) model. The overall dynamical behavior for κ>1 and κ=1 is quite different. The residual energy decays in a power law for both κ>1 and κ=1 from which the dynamical exponent z has been estimated. The persistence probability shows algebraic decay for κ>1 with an exponent θ=0.22±0.002 while the dynamical exponent for ordering z=2.33±0.01. For κ=1, the system belongs to a completely different dynamical class with θ=0.332±0.002 and z=2.47±0.04. We have computed the freezing probability for different values of κ. We have also studied the decay of autocorrelation function with time for a different regime of κ values. The results have been compared with those of the two-dimensional ANNNI model.
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.
The critical relaxation from the low-temperature ordered state of the three-dimensional Ising model with competing interactions on a simple cubic lattice has been studied for the first time using the short-time dynamics method. Competition between exchange interactions is due to the ferromagnetic interaction between the nearest neighbors and the antiferromagnetic interaction between the next nearest neighbors. Particles containing 262144 spins with periodic boundary conditions have been studied. Calculations have been performed by the standard Metropolis Monte Carlo algorithm. The static critical exponents of the magnetization and correlation radius have been calculated. The dynamic critical exponent of the model under study has been calculated for the first time.
The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.