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arXiv:2011.05567 · Physical review. E

Monte Carlo renormalization-group calculation for the d=3 Ising model using a modified transformation

Dorit Ron, Achi Brandt, Robert H. Swendsen

Published Aug 26, 2021• 3 citations
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Abstract

We present a simple approach to high-accuracy calculations of critical properties for the three-dimensional Ising model, without prior knowledge of the critical temperature. The iterative method uses a modified block-spin transformation with a tunable parameter to improve convergence in the Monte Carlo renormalization group trajectory. We found experimentally that the iterative method enables the calculation of the critical temperature simultaneously with a critical exponent.

DOI
10.1103/physreve.104.025311

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We present a simple approach to high-accuracy calculations of critical properties for the three-dimensional Ising model, without prior knowledge of the critical temperature. The iterative method uses a modified block-spin transformation with a tunable parameter to improve convergence in the Monte Carlo renormalization group trajectory. We found experimentally that the iterative method enables the calculation of the critical temperature simultaneously with a critical exponent.
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 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).
We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.