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 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 LettersDec 18, 200232 citations

Application of real-space renormalization group approach to critical dynamics

Gene F. Mazenko

Springer eBooksApr 6, 20080 citations

Renormalization Group Approach to Dynamic Critical Phenomena

Gene F. Mazenko

Springer eBooksJan 1, 19780 citations

Historical Review on Analytic, Monte Carlo, and Renormalization Group Approaches to Critical Phenomena of Some Lattice Models

Chin‐Kun Hu

Analytic calculations, Monte Carlo (MC) simulations [N. Metropolis and S. Ulam, J. Am. Stat. Asso 44, 335 (1949)] and renormalization group (RG) theory [K. G. Wilson, Phys. Rev. B 4, 3174 and 3184 (1971)] are important methods of statistical physics to study phase transitions and critical phenomena. This paper gives a brief historical review on analytic, MC, and RG approaches to some lattice critical systems in memory of late Professors Shang-keng Ma (1940-1983) and Yu-Ming Shih (1942-2005), who played some key roles in early developments of statistical physics of critical phenomena in Taiwan. The paper first introduces some developments in the study of critical phenomena from late 19th century to early 70s in the 20th century when renormalization group theory was formulated. The paper then reviews some topics on analytic, Monte Carlo, and RG approach to some lattice models, including variational RG approach to the Ising model, slow relaxation of a spin glass model at low temperatures, iterative method for quantum spin models, percolation theory of critical phenomena of Ising-like models and hard-core particle model, histogram Monte Carlo method and histogram Monte Carlo RG method, universal finite-size scaling functions of lattice models, critical slowing down of the Ising model, finite-size corrections for critical lattice models, and self-organized critical systems. Such topics are closely related to the author's own research experience. Finally, the paper outlines some very recent developments, summarizes main results of this paper, and points out some interesting problems for further studies, e.g. why a biological system can maintain in a non-equilibrium state for a very long time.Analytic calculations, Monte Carlo (MC) simulations [N. Metropolis and S. Ulam, J. Am. Stat. Asso 44, 335 (1949)] and renormalization group (RG) theory [K. G. Wilson, Phys. Rev. B 4, 3174 and 3184 (1971)] are important methods of statistical physics to study phase transitions and critical phenomena. This paper gives a brief historical review on analytic, MC, and RG approaches to some lattice critical systems in memory of late Professors Shang-keng Ma (1940-1983) and Yu-Ming Shih (1942-2005), who played some key roles in early developments of statistical physics of critical phenomena in Taiwan. The paper first introduces some developments in the study of critical phenomena from late 19th century to early 70s in the 20th century when renormalization group theory was formulated. The paper then reviews some topics on analytic, Monte Carlo, and RG approach to some lattice models, including variational RG approach to the Ising model, slow relaxation of a spin glass model at low temperatures, iterative method for quantum spin models, percolation theory of critical phenomena of Ising-like models and hard-core particle model, histogram Monte Carlo method and histogram Monte Carlo RG method, universal finite-size scaling functions of lattice models, critical slowing down of the Ising model, finite-size corrections for critical lattice models, and self-organized critical systems. Such topics are closely related to the author's own research experience. Finally, the paper outlines some very recent developments, summarizes main results of this paper, and points out some interesting problems for further studies, e.g. why a biological system can maintain in a non-equilibrium state for a very long time.

Chinese Journal of PhysicsFeb 1, 201446 citations

Lattice study of RG flow in 2D $\phi^4$ theoryLattice study of RG flow in 2D $\phi^4$ theory

Shashank Sharma, Dipankar Chakrabarti, Vipul Arora

Gradient flow method provides a practical framework for implementing Renormalization Group (RG) transformations on the lattice, commonly referred to as the Gradient Flow Renormalization Group (GFRG). In this work, we investigate the RG flow of the two-dimensional scalar $\phi^4$ theory in the vicinity of criticality. To address the problem of critical slowing down near the critical point, we employ Conditional Normalizing Flows (CNFs) to efficiently generate field configurations in the critical region. This combination of GFRG and CNFs enables a controlled non-perturbative study of critical behaviour and fixed-point structure.Gradient flow method provides a practical framework for implementing Renormalization Group (RG) transformations on the lattice, commonly referred to as the Gradient Flow Renormalization Group (GFRG). In this work, we investigate the RG flow of the two-dimensional scalar $\phi^4$ theory in the vicinity of criticality. To address the problem of critical slowing down near the critical point, we employ Conditional Normalizing Flows (CNFs) to efficiently generate field configurations in the critical region. This combination of GFRG and CNFs enables a controlled non-perturbative study of critical behaviour and fixed-point structure.

International Symposium on Lattice Field Theory (LATTICE)Jun 29, 20260 citations

Renormalization Group Approach to Critical Phenomena

H. Wagner

Springer eBooksJan 1, 19779 citations

Lattice study of RG flow in 2D

\phi^4

theory

\phi^4

theory in the vicinity of criticality. To address the problem of critical slowing down near the critical point, we employ Conditional Normalizing Flows (CNFs) to efficiently generate field configurations in the critical region. This combination of GFRG and CNFs enables a controlled non-perturbative study of critical behaviour and fixed-point structure.