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arXiv:cond-mat/9602020 · Physical Review Letters

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

Henk W. J. Blöte, Jouke R. Heringa, A. Hoogland, E. W. Meyer, T. S. Smit

Published Apr 8, 1996• 50 citations
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Abstract

We review the assumptions on which the Monte Carlo renormalization technique is based, in particular, the analyticity of the block-spin transformations. On this basis, we select an optimized Kadanoff blocking rule in combination with the simulation of a $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$ 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 ${y}_{T}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.585(3)$.

DOI
10.1103/physrevlett.76.2613

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We review the assumptions on which the Monte Carlo renormalization technique is based, in particular, the analyticity of the block-spin transformations. On this basis, we select an optimized Kadanoff blocking rule in combination with the simulation of a d=3d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3d=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 yH=2.481(1){y}_{H}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.481(1)yH​=2.481(1) and yT=1.585(3){y}_{T}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.585(3)yT​=1.585(3).
The majority-rule, block-spin, renormalization-group transformation for the d=3d=3d=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.
We test the performance of the Monte Carlo renormalization method in the context of the Ising model on a triangular lattice. We apply a block-spin transformation which allows for an adjustable parameter so that the transformation can be optimized. This optimization purportedly brings the fixed point of the transformation to a location where the corrections to scaling vanish. To this purpose we determine corrections to scaling of the triangular Ising model with nearest- and next-nearest-neighbor interactions by means of transfer-matrix calculations and finite-size scaling. We find that the leading correction to scaling just vanishes for the nearest-neighbor model. However, the fixed point of the commonly used majority-rule block-spin transformation appears to lie well away from the nearest-neighbor critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because the standard assumptions of real-space renormalization imply that corrections to scaling vanish at the fixed point. We avoid this inconsistency by means of the optimized transformation which shifts the fixed point back to the vicinity of the nearest-neighbor critical Hamiltonian. The results of the optimized transformation in terms of the Ising critical exponents are more accurate than those obtained with the majority rule.
A program combining the Monte Carlo technique and the Kadanoff block transformation is carried out for the Ising model in two and three dimensions, and the critical point and the critical indices $\ensuremath{\nu}$ and $\ensuremath{\delta}$ are obtained. A sequence of approximations seems to converge to the exact Onsager result in two dimensions, although the convergence of $\ensuremath{\nu}$ is rather slow. In three dimensions, the numbers obtained seem to tend towards the results from exact series expansions.
We present results of a Monte Carlo renormalization-group study of the three-dimensional Ising model on 643{64}^{3}643 and 1283{128}^{3}1283 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.