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Journal of Applied Physics

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

Rajan Gupta

Published Apr 15, 1987• 11 citations
Publisher

Abstract

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.

DOI
10.1063/1.338688

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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.
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.
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.
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.
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.