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

Journal of Anhui University of Technology

Monte Carlo Simulation of Real Space Renormalization

WU Da-yan

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

Mont Carlo simulation of the renormalization of 2-dimensional square lattice Ising model is performed, using simple sampling and important sampling, respectively. For simple sampling, analytical result is got, but only small cell can be simulated; Comparatively, the much larger cell can be simulated using. And the obtained numerical solution becomes closer to the accurate result. The method of this article can highly improve the accuracy of real space renormalization.

Related papers

Monte Carlo Renormalization Group and Ising Models with n>~2

Robert H. Swendsen, S. Krinsky

We suggest that the Monte Carlo renormalization group, when combined with the type of cell-spin transformation introduced by van Leeuwen, should be a powerful tool in the study of Ising models with $n>~2$. Numerical results are presented for the Baxter model and the Ising model with nearest- and next-nearest-neighbor interactions on a square lattice.

Physical Review Letters•Jul 16, 1979•145 citations

Monte Carlo renormalization group study of the two dimensional kinetic Ising model

John P. Bartel

The critical dynamics of the two dimensional Kinetic Ising Model have been studied using Monte Carlo Real Space Renormalization Group techniques. Time dependent correlation functions were calculated for square lattices of 16 x 16 and 32 x 32 spins, using periodic boundary conditions wth 60,000 and 15,000 Monte Carlo steps per spin, respectively;The critical exponent z was estimated using a matching process for a variety of renormalization group operators. The estimates for z were surprisingly sensitive to small changes in the value of the renormalization group operator, with physically reasonable operators giving estimates for z less than the rigorous lower bound for z of z(,o) = 1.75. The estimates for z from a given renormalization group operator showed poor convergence properties. Attempts to estimate z from the scaling form of the correlation function were inconclusive, as were attempts to estimate the static exponent (eta);Possible explanations of these unexpected results are discussed. The most plausible source of the complex behavior seen in this work is the discrete nature of time in the Monte Carlo computer simulation, which generates phenomena not present in standard theoretical treatments of the Kinetic Ising Model.

•Jan 1, 1984•0 citations

Real-space renormalization for Heisenberg models on two-dimensional lattices

V. O. Cheranovski, Thomas G. Schmalz, Daniel Klein

Real-space renormalization methods which simply shift and rescale the interaction parameters in the nearest-neighbor Heisenberg model are investigated for two-dimensional lattices of equivalent sites. Results are presented for the hexagonal, square-planar, triangular, and Kagome lattices via three different renormalization techniques. The first, which has been studied for some time, uses perturbation theory to evaluate the renormalized interactions. The second uses the variational method to improve on the perturbative results. The third method is based on a cluster expansion and is found to give much improved agreement with numerical results form Monte Carlo calculations, but it does not provide a variational bound to the exact solution. The dependence of all three methods on the size and shape of the renormalized block of sites is also investigated.

The Journal of Chemical Physics•Oct 1, 1994•11 citations

Monte Carlo mean-field method for spin systems

E. F. Henriques, Vera B. Henriques, Silvio R. A. Salinas

We use a Monte Carlo mean-field method proposed by Netz and Berker to analyze the critical behavior of an Ising square lattice. We show that this method demands longer sampling times as compared with the conventional Monte Carlo simulations. Also, similar mean-field results can be obtained from self-consistent analytic calculations for small clusters of spins.

Physical review. B, Condensed matter•Apr 1, 1995•6 citations

Critical Behavior Simulation of the Two-Dimensional Ising Model—A Computational Physics Study Based on the Monte Carlo Method

佳钢 李

To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.

Modern Physics•Jan 1, 2026•0 citations

Random Ising model in three dimensions: theory, experiment and simulation - a difficult coexistence

Bertrand Berche, Pierre Emmanuel Berche, Christophe Chatelain +1 more

We discuss different approaches to the study of the effect of disorder in the three-dimensional Ising model.From the theoretical point of view, renormalization group calculations provide quite accurate results.Experiments carried out on crystalline mixtures of compounds lead to measurements as accurate as three digits on the values of critical exponents.Numerically, extensive Monte Carlo simulations then pretend to be of comparable accuracy.Life becomes complicated when details are compared between the three approaches.

arXiv:cond-mat/0411255•Condensed Matter Physics•Jan 1, 2005•10 citations
Mont Carlo simulation of the renormalization of 2-dimensional square lattice Ising model is performed, using simple sampling and important sampling, respectively. For simple sampling, analytical result is got, but only small cell can be simulated; Comparatively, the much larger cell can be simulated using. And the obtained numerical solution becomes closer to the accurate result. The method of this article can highly improve the accuracy of real space renormalization.
We suggest that the Monte Carlo renormalization group, when combined with the type of cell-spin transformation introduced by van Leeuwen, should be a powerful tool in the study of Ising models with n> 2n>~2n> 2. Numerical results are presented for the Baxter model and the Ising model with nearest- and next-nearest-neighbor interactions on a square lattice.
The critical dynamics of the two dimensional Kinetic Ising Model have been studied using Monte Carlo Real Space Renormalization Group techniques. Time dependent correlation functions were calculated for square lattices of 16 x 16 and 32 x 32 spins, using periodic boundary conditions wth 60,000 and 15,000 Monte Carlo steps per spin, respectively;The critical exponent z was estimated using a matching process for a variety of renormalization group operators. The estimates for z were surprisingly sensitive to small changes in the value of the renormalization group operator, with physically reasonable operators giving estimates for z less than the rigorous lower bound for z of z(,o) = 1.75. The estimates for z from a given renormalization group operator showed poor convergence properties. Attempts to estimate z from the scaling form of the correlation function were inconclusive, as were attempts to estimate the static exponent (eta);Possible explanations of these unexpected results are discussed. The most plausible source of the complex behavior seen in this work is the discrete nature of time in the Monte Carlo computer simulation, which generates phenomena not present in standard theoretical treatments of the Kinetic Ising Model.
Real-space renormalization methods which simply shift and rescale the interaction parameters in the nearest-neighbor Heisenberg model are investigated for two-dimensional lattices of equivalent sites. Results are presented for the hexagonal, square-planar, triangular, and Kagome lattices via three different renormalization techniques. The first, which has been studied for some time, uses perturbation theory to evaluate the renormalized interactions. The second uses the variational method to improve on the perturbative results. The third method is based on a cluster expansion and is found to give much improved agreement with numerical results form Monte Carlo calculations, but it does not provide a variational bound to the exact solution. The dependence of all three methods on the size and shape of the renormalized block of sites is also investigated.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
We discuss different approaches to the study of the effect of disorder in the three-dimensional Ising model.From the theoretical point of view, renormalization group calculations provide quite accurate results.Experiments carried out on crystalline mixtures of compounds lead to measurements as accurate as three digits on the values of critical exponents.Numerically, extensive Monte Carlo simulations then pretend to be of comparable accuracy.Life becomes complicated when details are compared between the three approaches.