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Physical review. B, Condensed matter

Critical behavior of the three-dimensional Ising model: A high-resolution Monte Carlo study

Alan M. Ferrenberg, D. P. Landau

Published Sep 1, 1991• 786 citations
Publisher

Abstract

Using recently developed histogram techniques and an ultrafast multispin coding simulation algorithm, we have investigated the critical behavior of the d=3 simple-cubic Ising model. We have studied lattice sizes ranging from L=8 to 96 using between 3\ifmmode\times\else\texttimes\fi{}${10}^{6}$ and 12\ifmmode\times\else\texttimes\fi{}${10}^{6}$ Monte Carlo steps (complete lattice updates). By accurately measuring the finite-size behavior of several different thermodynamic quantities, we are able to determine the critical properties with a precision comparable to that obtained with Monte Carlo renormalization-group and sophisticated series-expansion techniques. The best estimate of the inverse critical temperature from our analysis is ${\mathit{K}}_{\mathit{c}}$=0.221 659 5\ifmmode\pm\else\textpm\fi{}0.000 002 6. The advantages of the histogram technique are discussed, as are the potential problems that can arise at this level of resolution.

DOI
10.1103/physrevb.44.5081

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Using recently developed histogram techniques and an ultrafast multispin coding simulation algorithm, we have investigated the critical behavior of the d=3 simple-cubic Ising model. We have studied lattice sizes ranging from L=8 to 96 using between 3\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 and 12\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 Monte Carlo steps (complete lattice updates). By accurately measuring the finite-size behavior of several different thermodynamic quantities, we are able to determine the critical properties with a precision comparable to that obtained with Monte Carlo renormalization-group and sophisticated series-expansion techniques. The best estimate of the inverse critical temperature from our analysis is Kc{\mathit{K}}_{\mathit{c}}Kc​=0.221 659 5\ifmmode\pm\else\textpm\fi{}0.000 002 6. The advantages of the histogram technique are discussed, as are the potential problems that can arise at this level of resolution.
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.
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).
The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with 83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643 sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling K1c=0.221654(6){K}_{1}^{c}=0.221654(6)K1c​=0.221654(6). After allowing for (i) interpolation to this best estimate for K1c{K}_{1}^{c}K1c​, (ii) an apparent finite-size effect in the renormalization-group transformation due to the measurement of correlation functions of too few (seven) operators, and (iii) the extrapolation for the effect of a slow transient towards the fixed point, the values $\ensuremath{\nu}=0.629(4)$ and $\ensuremath{\eta}=0.031(5)$ are obtained for the thermal and magnetic exponents. The correction-to-scaling exponent $\ensuremath{\omega}$ is estimated to be around 1; to obtain an accuracy competitive with other methods requires measurements with more than seven operators. We briefly review the problem of redundant operators and indicate the future prospects for this kind of calculation.
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.
The static critical properties of classical, isotropic Heisenberg ferromagnets have been studied numerically. Using an improved cluster algorithm we performed extensive Monte Carlo simulations on L×L×L simple cubic and body-centered cubic systems with L≤40 over a temperature range covering the critical region. Temperature dependences of thermodynamic quantities were determined with an optimized multiple-histogram method, and both critical temperatures and critical exponents were extracted from finite-size scaling of several different thermodynamic quantities. It was found that within their respective error bars the critical exponents for both ferromagnets agree with each other and with field theoretical predictions.