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

Physical Review Letters

Optimization of Real-Space Renormalization-Group Transformations

Robert H. Swendsen

Published Jun 25, 1984• 53 citations
Publisher

Abstract

A general method for optimizing real-space renormalization-group transformations is presented. It is applicable to both spin systems and lattice gauge theories, with use of Monte Carlo computer simulations. It provides substantial improvement in the convergence of critical exponents for the $d=3$ Ising model.

DOI
10.1103/physrevlett.52.2321

Related papers

Monte Carlo Calculation of Renormalized Coupling Parameters

Robert H. Swendsen

A practical method is presented for calculating many-parameter renormalization-group trajectories for both spin systems and lattice gauge theories from Monte Carlo computer simulations. It provides a direct check on the assumption of short-range renormalized interactions, and preliminary work indicates improvement of the convergence in the calculation of critical exponents.

Physical Review Letters•Apr 2, 1984•115 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

Optimal Real-Space Renormalization-Group Transformations with Artificial Neural Networks

Jui-Hui Chung, Ying-Jer Kao

We introduce a general method for optimizing real-space renormalization-group transformations to study the critical properties of a classical system. The scheme is based on minimizing the Kullback-Leibler divergence between the distribution of the system and the normalized normalizing factor of the transformation parametrized by a restricted Boltzmann machine. We compute the thermal critical exponent of the two-dimensional Ising model using the trained optimal projector and obtain a very accurate thermal critical exponent $y_t=1.0001(11)$ after the first step of the transformation.

arXiv:1912.09005•Dec 19, 2019•0 citations

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

Dorit Ron, Achi Brandt, Robert H. Swendsen

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.

arXiv:1703.02430•Physical review. E•May 16, 2017•21 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 general method for optimizing real-space renormalization-group transformations is presented. It is applicable to both spin systems and lattice gauge theories, with use of Monte Carlo computer simulations. It provides substantial improvement in the convergence of critical exponents for the d=3d=3d=3 Ising model.
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 introduce a general method for optimizing real-space renormalization-group transformations to study the critical properties of a classical system. The scheme is based on minimizing the Kullback-Leibler divergence between the distribution of the system and the normalized normalizing factor of the transformation parametrized by a restricted Boltzmann machine. We compute the thermal critical exponent of the two-dimensional Ising model using the trained optimal projector and obtain a very accurate thermal critical exponent yt=1.0001(11)y_t=1.0001(11)yt​=1.0001(11) after the first step of the transformation.
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).