Physical review. B, Condensed matter

Monte Carlo calculation of renormalized coupling parameters. II. d=3 Ising model

Robert H. Swendsen

Published Oct 1, 1984• 32 citations

Abstract

The majority-rule, block-spin, renormalization-group transformation for the $d=3$ Ising model is studied, with the use of a recently developed technique for calculating renormalized coupling parameters from a single Monte Carlo simulation. Renormalization-group trajectories and the approximate fixed point are calculated. Improvements in the convergence of the Monte Carlo renormalization-group calculations of critical exponents are discussed.

DOI: 10.1103/physrevb.30.3875

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Physical review. B, Condensed matter

Monte Carlo calculation of renormalized coupling parameters. II. d=3 Ising model

Robert H. Swendsen

Published Oct 1, 1984• 32 citations

Publisher

Abstract

The majority-rule, block-spin, renormalization-group transformation for the

d=3

Ising model is studied, with the use of a recently developed technique for calculating renormalized coupling parameters from a single Monte Carlo simulation. Renormalization-group trajectories and the approximate fixed point are calculated. Improvements in the convergence of the Monte Carlo renormalization-group calculations of critical exponents are discussed.

DOI: 10.1103/physrevb.30.3875

Related papers

Monte Carlo calculation of renormalized coupling parameters. I. d=2 Ising model

Robert H. Swendsen

Renormalization-group trajectories produced by various real-space, block-spin transformations of the $d=2$ Ising model are studied, using a recently developed technique for calculating renormalized coupling parameters from a Monte Carlo simulation of a single system.Renormalization-group trajectories produced by various real-space, block-spin transformations of the $d=2$ Ising model are studied, using a recently developed technique for calculating renormalized coupling parameters from a Monte Carlo simulation of a single system.

Physical review. B, Condensed matterOct 1, 198451 citations

Monte Carlo Renormalization of the 3D Ising Model: Analyticity and Convergence

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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.

Physical review. B, Condensed matterApr 1, 1984318 citations

The majority-rule, block-spin, renormalization-group transformation for the

d=3

Renormalization-group trajectories produced by various real-space, block-spin transformations of the

d=2

Ising model are studied, using a recently developed technique for calculating renormalized coupling parameters from a Monte Carlo simulation of a single system.

d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3

Ising model with reduced corrections to scaling. This is achieved by including interactions with second and third neighbors. As a consequence of the improved transformation, this Monte Carlo renormalization method yields a fast convergence and a high accuracy. The results for the critical exponents are

{y}_{H}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.481(1)

and

{y}_{T}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.585(3)

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.