Open problems in Monte Carlo Renormalization Group application to critical phenomenon

Rajan Gupta

Published Jan 1, 1986• 0 citations

Abstract

The Monte Carlo Renormalization Group (MCRG) methods and the theory behind them are reviewed. The Gupta-Cordery improved MCRG method is described and compared with the standard one. The emphasis is on the progress made in understanding the truncation errors in the Linearized Transformation Matrix and on open problems. Lastly, some of the existing methods for calculating the renormalized Hamiltonian are reviewed and evaluated. 35 refs.

Related papers

Open problems in Monte Carlo renormalization group: Application to critical phenomena (invited)

Rajan Gupta

The Monte Carlo renormalization group (MCRG) methods and the underlying theory is reviewed. The Gupta–Cordery improved MCRG method is described and compared with the standard one. The emphasis is on the progress made in understanding the truncation errors in the linearized transformation matrix and on open problems. The problem of deteriorating exponents with long-range transformations is highlighted. Lastly, some of the existing methods for calculating the renormalized Hamiltonian are reviewed and evaluated.

Journal of Applied PhysicsApr 15, 198711 citations

MONTE CARLO RENORMALIZATION GROUP STUDY OF THE D=1 XXZ MODEL

M. A. Novotny, Hans Gerd Evertz

We report current progress on the synthesis of methods to alleviate two major difficulties in implementing a Monte Carlo Renormalization Group (MCRG) for quantum systems. In particular, we have utilized the loop-algorithm to reduce critical slowing down, and we have implemented an MCRG method in which the symmetries of the classical equivalent model need not be fully understood, since the Renormalization Group is given by the Monte Carlo simulation. We report preliminary results obtained when the resulting MCRG method is applied to the d=1 XXZ model. Our results are encouraging. However, since this model has a Kosterlitz-Thouless transition, it does not yet provide a full test of our MCRG method.

arXiv:cond-mat/9309047WORLD SCIENTIFIC eBooksDec 1, 19931 citation

Monte Carlo renormalization-group studies of critical phenomena

Robert H. Swendsen

Monte Carlo renormalization-group (MCRG) methods are described and illustrated for models of current interest. The calculation of critical exponents and the determination of critical temperatures are discussed, as well as the application to multicritical points.

Journal of Applied PhysicsMar 1, 198210 citations

Monte Carlo renormalization group for quantum systems

M. A. Novotny, David Paul Landau

A Monte Carlo renormalization-group (MCRG) method developed to study quantum systems is presented. This MCRG method is the synthesis of the Monte Carlo method for quantum systems using the Trotter transformation with the MCRG developed for classical systems. It is applicable to the study of either the ground-state or the finite-temperature behavior of a quantum system. The convergence properties of this MCRG are illustrated by applying it to the ground state of the d=1 Ising model in a transverse magnetic field. Very rapid convergence to the known critical exponents is exhibited even for small lattice sizes.

Physical review. B, Condensed matterFeb 1, 19859 citations

Critical behavior of the three-state Potts model: Monte Carlo renormalization group

Robert H. Swendsen, A. Nihat Berker

The critical behavior of the three-state Potts model in two dimensions is investigated with the use of the Monte Carlo renormalization group (MCRG), and the results are compared with conjectured values as well as with other methods. Recent advances in the MCRG enable greatly improved convergence, as well as greater insight into the structure of the renormalization transformation. The introduction of a four-point interaction is used to control the chemical potential of effective vacancies in a purely Potts Hamiltonian.

Physical review. B, Condensed matterOct 1, 19837 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.

Chinese Journal of PhysicsFeb 1, 201446 citations