A new MCRG (Monte Carlo Renormalization Group) calculation of the critical behavior of the 3d ising model - preliminary results

CLIVE F. BAILLIE, Kenneth N. Barish, Rajan Gupta, Godfrey Stuart Pawley

Published Oct 5, 1989• 0 citations

Abstract

We have undertaken a large-scale computer simulation of the 3d Ising model using the Monte Carlo Renormalization Group (MCRG) method. Our calculation improves on past calculations in three ways. Firstly, larger lattices are used -- we have run on 128{sup 3} as well as 64{sup 3}. Secondly, many more spin operators -- 53 even and 46 odd -- are measured. Lastly, we have incorporated more efficient ways of spin update -- Wolff's single-cluster variant of the Swendsen and Wang algorithm. In addition to calculating the critical temperature the thermal and magnetic exponents, and improving the estimate of the correction-to-scaling exponent, we are aiming to verify the presence of redundant operators. We do not yet have sufficient statistics to give final results so here we describe our calculation and present preliminary results which demonstrate that we have control over both finite size effects and systematic errors arising from truncation. 16 refs., 7 figs.

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A new MCRG (Monte Carlo Renormalization Group) calculation of the critical behavior of the 3d ising model - preliminary results

CLIVE F. BAILLIE, Kenneth N. Barish, Rajan Gupta, Godfrey Stuart Pawley

Published Oct 5, 1989• 0 citations

OpenAlex

Abstract

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Simulations of the 3D Ising Model at Criticality

Henk W. J. Blöte, G. Kamieniarz

Acta Physica Polonica AFeb 1, 19946 citations

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.

{100}^{3}

spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.