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Physical review. E

Monte Carlo static and dynamic simulations of a three-dimensional Ising critical model

F. Livet

Published Feb 27, 2020• 1 citation
Publisher

Abstract

The critical dynamics of `model A'' of Hohenberg and Halperin has been studied by the Monte Carlo method. Simulations have been carried out in the three-dimensional (3d) simple cubic Ising model for lattices of sizes $L=16$ to $L=512$. Using Wolff's cluster algorithm, the critical temperature is precisely found as ${\ensuremath{\beta}}_{c}=0.221\phantom{\rule{0.16em}{0ex}}654\phantom{\rule{0.16em}{0ex}}68(5)$. By Fourier transform of the lattice configurations, the critical scattering intensities $I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ can be obtained. After circular averaging, the static simulations with $L=256$ and $L=512$ provide an estimate of the critical exponent $\ensuremath{\gamma}/\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\eta}=1.9640(5)$. The $|\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|$-dependent distribution of $I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ showed an exponential distribution, corresponding to a Gaussian distribution of the scattering amplitudes for a large $q$ domain. The time-dependent intensities were then used for the study of the critical dynamics of 3d lattices at the critical point. To simulate results of an x-ray photon correlation spectroscopy experiment, the time-dependent correlation function of the intensities was studied for each $|\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|$-value. In the $q$ region where $I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ had an exponential distribution, the time correlations can be fit to a stretched exponential, where the exponent $\ensuremath{\mu}=\ensuremath{\gamma}/\ensuremath{\nu}z\ensuremath{\simeq}0.975$ provides an estimate of the dynamic exponent $z$. This corresponds to $z=2.0145$, in agreement with the observed variations of the characteristic fluctuation time of the intensity: $\ensuremath{\tau}(q)\ensuremath{\propto}{q}^{\ensuremath{-}z}$, which gives $z=2.015$. These results agree with the $\ensuremath{\epsilon}$ expansion of field-theoretical methods (2.017). In this paper, the need to take account of the anomalous time behavior ($\ensuremath{\mu}<1$) in the dynamics is exemplified. This dynamics reflects a nonlinear time behavior of model A, and its large time extension is discussed in detail.

DOI
10.1103/physreve.101.022131

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The critical dynamics of `model A'' of Hohenberg and Halperin has been studied by the Monte Carlo method. Simulations have been carried out in the three-dimensional (3d) simple cubic Ising model for lattices of sizes L=16L=16L=16 to L=512L=512L=512. Using Wolff's cluster algorithm, the critical temperature is precisely found as \ensuremathβc=0.22165468(5){\ensuremath{\beta}}_{c}=0.221\phantom{\rule{0.16em}{0ex}}654\phantom{\rule{0.16em}{0ex}}68(5)\ensuremathβc​=0.22165468(5). By Fourier transform of the lattice configurations, the critical scattering intensities I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}) can be obtained. After circular averaging, the static simulations with L=256L=256L=256 and L=512L=512L=512 provide an estimate of the critical exponent \ensuremathγ/\ensuremathν=2\ensuremath−\ensuremathη=1.9640(5)\ensuremath{\gamma}/\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\eta}=1.9640(5)\ensuremathγ/\ensuremathν=2\ensuremath−\ensuremathη=1.9640(5). The |\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|-dependent distribution of I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}) showed an exponential distribution, corresponding to a Gaussian distribution of the scattering amplitudes for a large qqq domain. The time-dependent intensities were then used for the study of the critical dynamics of 3d lattices at the critical point. To simulate results of an x-ray photon correlation spectroscopy experiment, the time-dependent correlation function of the intensities was studied for each |\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|-value. In the qqq region where I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}) had an exponential distribution, the time correlations can be fit to a stretched exponential, where the exponent \ensuremathμ=\ensuremathγ/\ensuremathνz\ensuremath≃0.975\ensuremath{\mu}=\ensuremath{\gamma}/\ensuremath{\nu}z\ensuremath{\simeq}0.975\ensuremathμ=\ensuremathγ/\ensuremathνz\ensuremath≃0.975 provides an estimate of the dynamic exponent zzz. This corresponds to z=2.0145z=2.0145z=2.0145, in agreement with the observed variations of the characteristic fluctuation time of the intensity: \ensuremathτ(q)\ensuremath∝q\ensuremath−z\ensuremath{\tau}(q)\ensuremath{\propto}{q}^{\ensuremath{-}z}\ensuremathτ(q)\ensuremath∝q\ensuremath−z, which gives z=2.015z=2.015z=2.015. These results agree with the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion of field-theoretical methods (2.017). In this paper, the need to take account of the anomalous time behavior (\ensuremathμ<1\ensuremath{\mu}<1\ensuremathμ<1) in the dynamics is exemplified. This dynamics reflects a nonlinear time behavior of model A, and its large time extension is discussed in detail.
Using both recently developed cluster-algorithm and histogram methods, we have carried out a high-resolution Monte Carlo study of static critical properties of classical ferromagnetic Heisenberg models. Extensive Monte Carlo simulations were performed at several temperatures in the critical region, using an improved cluster-updating scheme, on L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic and body-centered-cubic systems with L\ensuremath{\le}40. Thermodynamic quantities as a function of temperature in the vicinity of the critical point were obtained by an optimized multiple-histogram method, and the critical temperature and static critical exponents were extracted using finite-size scaling. Our best estimates for the inverse critical temperatures are 0.693 035(37) for the simple-cubic system and 0.486 798(12) for the body-centered-cubic system. Estimated static critical exponents for both systems agree with each other within their respective error bars, and the mean estimates \ensuremath{\nu}=0.7048(30) and \ensuremath{\gamma}=1.3873(85) are also in excellent agreement with field-theoretic predictions 0.705(3) and 1.386(4).
In this work we performed numerical simulations for the Ising model on three dimensional lattices with coordination number equal 5. With Monte Carlo simulations in the static case we evaluated the critical temperature and the static critical exponents ννν, γγγ and βββ. Once that we have the critical temperature value we investigated the dynamical critical behavior with Glauber dynamics, starting from disordered states. From our simulations we obtained the values for the dynamic exponent z=2.037(8)z=2.037(8)z=2.037(8), the exponent for the autocorrelation λ/z=1.364(5)λ/z=1.364(5)λ/z=1.364(5), the exponent of the critical initial increase θ′=0.109(8)θ'=0.109(8)θ′=0.109(8) and the asymptotic value of the fluctuation-dissipation ratio X∞=0.433(1)X^\infty=0.433(1)X∞=0.433(1). All of these results are in good agreement to the previous values reported for the 3D Ising model universality class.
We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}1012{10}^{12}1012 Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.
While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.
Using extensive Monte Carlo simulations that employ the Wolff cluster flipping and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising model with lattice sizes ranging from 16 3 to 1024 3 . By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, we obtained the critical inverse temperature K c = 0.221 654 626(5) and the critical exponent of the correlation length ν = 0.629 912(86) with precision that improves upon previous Monte Carlo estimates.
We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size LLL: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter ppp, p=0.5p=0.5p=0.5 and p=0.7p=0.7p=0.7 (up to L=28L=28L=28 and L=20L=20L=20, respectively), and the bond-diluted bimodal model for bond-occupation probability pb=0.45{p}_{b}=0.45pb​=0.45 (up to L=16L=16L=16). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at pb=0.7{p}_{b}=0.7pb​=0.7 and pb=0.35{p}_{b}=0.35pb​=0.35 (up to L=10L=10L=10) and of the Ising spin glass model with Gaussian bond distribution (up to L=8L=8L=8) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.