Journal of Physics A Mathematical and General

Treatment of the 3D Ising gauge model by a Monte Carlo renormalisation group method

Stephan Wansleben

Published Mar 11, 1985• 2 citations

Abstract

The Ising gauge model on a simple cubic lattice with finite extension in one direction is treated by Binder's Monte Carlo renormalisation group method (1981). beta / nu is found to increase with increasing number of layers to values larger than 1/8.

DOI: 10.1088/0305-4470/18/4/006

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Journal of Physics A Mathematical and General

Treatment of the 3D Ising gauge model by a Monte Carlo renormalisation group method

Stephan Wansleben

Published Mar 11, 1985• 2 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/18/4/006

Related papers

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Melanie Schulte, Caroline Drope

arXiv:cond-mat/0501731Jan 29, 20050 citations

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Melanie Schulte, Caroline Drope

International Journal of Modern Physics CAug 1, 200512 citations

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.

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.