Physical review. B, Condensed matter

Critical behavior of the three-dimensional Ising model

Henk W. J. Blöte, Robert H. Swendsen

Published Sep 1, 1979• 44 citations

Abstract

The simple-cubic Ising model has been investigated with the Monte Carlo renormalization-group method. The results are found to be consistent with the critical exponents obtained from series and $\ensuremath{\epsilon}$ expansions.

DOI: 10.1103/physrevb.20.2077

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

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Physical review. B, Condensed matter

Critical behavior of the three-dimensional Ising model

Henk W. J. Blöte, Robert H. Swendsen

Published Sep 1, 1979• 44 citations

Publisher

Abstract

The simple-cubic Ising model has been investigated with the Monte Carlo renormalization-group method. The results are found to be consistent with the critical exponents obtained from series and

\ensuremath{\epsilon}

expansions.

DOI: 10.1103/physrevb.20.2077

Related papers

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

Alan M. Ferrenberg, D. P. Landau

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.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.

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Monte Carlo renormalization-group calculations of critical behavior in the simple-cubic Ising model

Godfrey Stuart Pawley, Robert H. Swendsen, D. J. Wallace +1 more

The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with ${8}^{3}$ , ${16}^{3}$ , ${32}^{3}$ , and ${64}^{3}$ sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling ${K}_{1}^{c}=0.221654(6)$ . After allowing for (i) interpolation to this best estimate for ${K}_{1}^{c}$ , (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.The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with ${8}^{3}$, ${16}^{3}$, ${32}^{3}$, and ${64}^{3}$ sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling ${K}_{1}^{c}=0.221654(6)$. After allowing for (i) interpolation to this best estimate for ${K}_{1}^{c}$, (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.

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Monte Carlo renormalization-group study of the three-dimensional Ising model

CLIVE F. BAILLIE, Rajan Gupta, Ken A. Hawick +1 more

We present results of a Monte Carlo renormalization-group study of the three-dimensional Ising model on ${64}^{3}$ and ${128}^{3}$ simple-cubic lattices. The eigenvalues of a linearized transformation matrix, constructed with the use of 53 even operators and 46 odd operators, are shown to be free of truncation errors. Our estimate of the critical coupling is 0.221 652\ifmmode\pm\else\textpm\fi{}0.000 003\ifmmode\pm\else\textpm\fi{}0.000 001 where the first error is statistical and the second due to the finite number of blocking steps. The results for the relevant exponents are \ensuremath{\nu}=0.624(2) and \ensuremath{\eta}=0.026(3). This estimate for \ensuremath{\nu} lies (2--3)\ensuremath{\sigma} below that obtained from other methods. The correction-to-scaling exponent is found to lie in the range \ensuremath{\omega}=0.8--0.85. We also find that the subleading magnetic exponent is relevant and present evidence that it corresponds to a redundant eigenvector of the majority-rule blocking transformation.We present results of a Monte Carlo renormalization-group study of the three-dimensional Ising model on ${64}^{3}$ and ${128}^{3}$ simple-cubic lattices. The eigenvalues of a linearized transformation matrix, constructed with the use of 53 even operators and 46 odd operators, are shown to be free of truncation errors. Our estimate of the critical coupling is 0.221 652\ifmmode\pm\else\textpm\fi{}0.000 003\ifmmode\pm\else\textpm\fi{}0.000 001 where the first error is statistical and the second due to the finite number of blocking steps. The results for the relevant exponents are \ensuremath{\nu}=0.624(2) and \ensuremath{\eta}=0.026(3). This estimate for \ensuremath{\nu} lies (2--3)\ensuremath{\sigma} below that obtained from other methods. The correction-to-scaling exponent is found to lie in the range \ensuremath{\omega}=0.8--0.85. We also find that the subleading magnetic exponent is relevant and present evidence that it corresponds to a redundant eigenvector of the majority-rule blocking transformation.

Physical review. B, Condensed matterMay 1, 1992174 citations

MCRG study of d-dimensional random field Ising models

Elisabeth Busk Rasmussen, M. A. Novotny, David Paul Landau

Journal of Applied PhysicsMar 1, 198231 citations

Critical behavior of the four-dimensional Ising model

Henk W. J. Blöte, Robert H. Swendsen

Physical review. B, Condensed matterNov 1, 198037 citations

The simple-cubic Ising model has been investigated with the Monte Carlo renormalization-group method. The results are found to be consistent with the critical exponents obtained from series and

\ensuremath{\epsilon}

expansions.

{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.

The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with

{8}^{3}

{16}^{3}

{32}^{3}

, and

{64}^{3}

sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling

{K}_{1}^{c}=0.221654(6)

. After allowing for (i) interpolation to this best estimate for

{K}_{1}^{c}

, (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 present results of a Monte Carlo renormalization-group study of the three-dimensional Ising model on

{64}^{3}

and

{128}^{3}

simple-cubic lattices. The eigenvalues of a linearized transformation matrix, constructed with the use of 53 even operators and 46 odd operators, are shown to be free of truncation errors. Our estimate of the critical coupling is 0.221 652\ifmmode\pm\else\textpm\fi{}0.000 003\ifmmode\pm\else\textpm\fi{}0.000 001 where the first error is statistical and the second due to the finite number of blocking steps. The results for the relevant exponents are \ensuremath{\nu}=0.624(2) and \ensuremath{\eta}=0.026(3). This estimate for \ensuremath{\nu} lies (2--3)\ensuremath{\sigma} below that obtained from other methods. The correction-to-scaling exponent is found to lie in the range \ensuremath{\omega}=0.8--0.85. We also find that the subleading magnetic exponent is relevant and present evidence that it corresponds to a redundant eigenvector of the majority-rule blocking transformation.