Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1703.02430 · Physical review. E

Surprising convergence of the Monte Carlo renormalization group for the three-dimensional Ising model

Dorit Ron, Achi Brandt, Robert H. Swendsen

Published May 16, 2017• 21 citations
arXivHTMLPDF

Abstract

We present a surprisingly simple approach to high-accuracy calculations of the critical properties of the three-dimensional Ising model. The method uses a modified block-spin transformation with a tunable parameter to improve convergence in the Monte Carlo renormalization group. The block-spin parameter must be tuned differently for different exponents to produce optimal convergence.

DOI
10.1103/physreve.95.053305

Related papers

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

Dorit Ron, Achi Brandt, Robert H. Swendsen

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.

arXiv:2011.05567•Physical review. E•Aug 26, 2021•3 citations

Monte Carlo calculation of renormalized coupling parameters. II. d=3 Ising model

Robert H. Swendsen

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.

Physical review. B, Condensed matter•Oct 1, 1984•32 citations

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

Henk W. J. Blöte, Jouke R. Heringa, A. Hoogland +2 more

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 ${y}_{T}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.585(3)$.

arXiv:cond-mat/9602020•Physical Review Letters•Apr 8, 1996•50 citations

A Simple Method of Monte Carlo Renormalization Group

Macoto Kikuchi, Yutaka Okabe

We propose a simple method of Monte Carlo renormalization group which is based on the transformation properties of the thermodynamic quantities under the block-spin transformation. We apply the method to the three-dimensional Ising model.

Progress of Theoretical Physics•Jan 1, 1986•16 citations

Inverse Monte Carlo Renormalization Group Transformations for Critical Phenomena

Dorit Ron, Robert H. Swendsen, Achi Brandt

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.

Physical Review Letters•Dec 18, 2002•32 citations

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.

Progress of Theoretical Physics•Sep 1, 1987•10 citations
We present a surprisingly simple approach to high-accuracy calculations of the critical properties of the three-dimensional Ising model. The method uses a modified block-spin transformation with a tunable parameter to improve convergence in the Monte Carlo renormalization group. The block-spin parameter must be tuned differently for different exponents to produce optimal convergence.
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.
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.