Physical Review Letters

Critical Properties from Monte Carlo Coarse Graining and Renormalization

Kurt Binder

Published Aug 31, 1981• 801 citations

Abstract

The distribution function ${P}_{L}(s)$ of the local order parameter $s$ in finite blocks of size ${L}^{d}$ is studied for Ising models for dimensionalities $d=2, 3, \mathrm{and} 4$ by Monte Carlo methods. A real-space renormalization group based on phenomenological scaling yields fairly accurate results for rather small $L$ (e.g., the standard exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$ for $d=3$ are found as $\frac{2\ensuremath{\beta}}{\ensuremath{\nu}}=1.03\ifmmode\pm\else\textpm\fi{}0.01$, $\frac{1}{\ensuremath{\nu}}=1.60\ifmmode\pm\else\textpm\fi{}0.05$). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities $n>1$.

DOI: 10.1103/physrevlett.47.693

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Physical Review Letters

Critical Properties from Monte Carlo Coarse Graining and Renormalization

Kurt Binder

Published Aug 31, 1981• 801 citations

Publisher

Abstract

The distribution function

{P}_{L}(s)

of the local order parameter

s

in finite blocks of size

{L}^{d}

is studied for Ising models for dimensionalities

d=2, 3, \mathrm{and} 4

by Monte Carlo methods. A real-space renormalization group based on phenomenological scaling yields fairly accurate results for rather small

L

(e.g., the standard exponents

\ensuremath{\beta}

and

\ensuremath{\nu}

for

d=3

are found as

\frac{2\ensuremath{\beta}}{\ensuremath{\nu}}=1.03\ifmmode\pm\else\textpm\fi{}0.01

\frac{1}{\ensuremath{\nu}}=1.60\ifmmode\pm\else\textpm\fi{}0.05

). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities

n>1

DOI: 10.1103/physrevlett.47.693

Related papers

A Scaling Approach to Monte Carlo Renormalization Group

Macoto Kikuchi, Yutaka Okabe

We propose a simple method of Monte Carlo renormalization group. Thermodynamic quantities of the block-spin system are analyzed phenomenologically on the basis of scaling concept. The field exponent yH is obtained from the square of block-spin magnetization at the fixed point, while the temperature exponent yT is obtained as a consequence of nonlinearity of the scaling fields. In order to check the effectiveness of the method, we perform calculation on the three-dimensional Ising model. The critical temperature and the two critical exponents are estimated precisely. Finite-size effect is shown to be weak, whose origin is the regular part of the free energy density. Statistical errors of the ratios of the block-spin quantities are proved to be surprisingly small compared to those of the block-spin quantities themselves.We propose a simple method of Monte Carlo renormalization group. Thermodynamic quantities of the block-spin system are analyzed phenomenologically on the basis of scaling concept. The field exponent yH is obtained from the square of block-spin magnetization at the fixed point, while the temperature exponent yT is obtained as a consequence of nonlinearity of the scaling fields. In order to check the effectiveness of the method, we perform calculation on the three-dimensional Ising model. The critical temperature and the two critical exponents are estimated precisely. Finite-size effect is shown to be weak, whose origin is the regular part of the free energy density. Statistical errors of the ratios of the block-spin quantities are proved to be surprisingly small compared to those of the block-spin quantities themselves.

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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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The distribution function

{P}_{L}(s)

of the local order parameter

s

in finite blocks of size

{L}^{d}

is studied for Ising models for dimensionalities

d=2, 3, \mathrm{and} 4

by Monte Carlo methods. A real-space renormalization group based on phenomenological scaling yields fairly accurate results for rather small

L

(e.g., the standard exponents

\ensuremath{\beta}

and

\ensuremath{\nu}

for

d=3

are found as

\frac{2\ensuremath{\beta}}{\ensuremath{\nu}}=1.03\ifmmode\pm\else\textpm\fi{}0.01

\frac{1}{\ensuremath{\nu}}=1.60\ifmmode\pm\else\textpm\fi{}0.05

). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities

n>1

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.

L

, towards scaled universal forms. Estimates of the order parameter and susceptibility are obtained from the distribution functions. The critical temperature is estimated by a method independent of the estimate of the critical exponents. The critical exponent $\ensuremath{\nu}$ is obtained with the use of finite-size scaling arguments. The joint distribution function shows a double-well structure for

T<{T}_{c}

and can be represented by a coarse-grained effective Hamiltonian of two cell variables. We have determined the dependence of this functional on the coarse-graining cell size for several temperatures. In particular, we have calculated the dependence of the "spinodal curve" on the (scaled) coarse-graining size. The relevance of this work to the kinetics of first-order phase transitions is also discussed.

We present a real-space renormalization-group (RG) study of the

n

-component classical Heisenberg model in the large-

n

limit. We obtain exact expressions for correlation functions of block spins, given those of the original spins. We show how to extract the critical behavior of the model from such information. The model provides a good testing ground for evaluating the effectiveness of Monte Carlo RG techniques. We study numerically the implementation of the real-space RG transformation on finite lattices. In two dimensions, we obtain the $\ensuremath{\beta}$ function as a function of temperature. In three dimensions, we calculate the critical temperature and the thermal exponent. We discuss the effects of the use of finite lattices and of the use of different forms for the initial action on the accuracy of the results.

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.