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Physical review. B, Condensed matter

Reweighting in Monte Carlo and Monte Carlo renormalization-group studies

E. Peter Münger, M. A. Novotny

Published Mar 1, 1991• 34 citations

Abstract

A method of reweighting Monte Carlo data to parameters other than the simulated ones is investigated. The method is also extended to include Monte Carlo renormalization-group simulations. Both single and multiple simulation reweighting is used. The models studied are the d=2 ferromagnetic q=3 Potts model and the d=2 antiferromagnetic Ising model. It is shown that the reweighting produces a systematic shift in the height and location of the specific-heat peak. This shift, which without proper care may lead to erroneous results, depends on the simulated and reweighted temperatures and on the number of configurations included in the reweighting. Only for very large numbers of configurations does the shift decrease as the inverse square root of the number of configurations. Within the Monte Carlo renormalization-group study, it is shown that the critical exponents for the Ising model are determined more accurately from a multiple reweighting using one simulation below the critical temperature and one above than from the individual simulations. It is shown that it is more efficient to use two simulations at a time than all of them at once when reweighting multiple simulations.

DOI: 10.1103/physrevb.43.5773

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A method of reweighting Monte Carlo data to parameters other than the simulated ones is investigated. The method is also extended to include Monte Carlo renormalization-group simulations. Both single and multiple simulation reweighting is used. The models studied are the d=2 ferromagnetic q=3 Potts model and the d=2 antiferromagnetic Ising model. It is shown that the reweighting produces a systematic shift in the height and location of the specific-heat peak. This shift, which without proper care may lead to erroneous results, depends on the simulated and reweighted temperatures and on the number of configurations included in the reweighting. Only for very large numbers of configurations does the shift decrease as the inverse square root of the number of configurations. Within the Monte Carlo renormalization-group study, it is shown that the critical exponents for the Ising model are determined more accurately from a multiple reweighting using one simulation below the critical temperature and one above than from the individual simulations. It is shown that it is more efficient to use two simulations at a time than all of them at once when reweighting multiple simulations.

We combine histogram reweighting techniques with the two-lattice matching Monte Carlo renormalization group method to conduct computationally efficient calculations of critical exponents on systems with moderately small lattice sizes. The approach, which relies on the construction of renormalization group mappings between two systems of identical lattice size to partially eliminate finite-size effects, and the use of histogram reweighting to obtain computationally efficient results in extended regions of parameter space, is utilized to explicitly determine the renormalized coupling parameters of the two-dimensional ϕ^{4} scalar field theory and to extract multiple critical exponents. We conclude by quantifying the computational benefits of the approach and discuss how reweighting opens up the opportunity to extend Monte Carlo renormalization group methods to systems with complex-valued actions.

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.

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.

We present an iterative Monte Carlo algorithm for which the temperature variable is attracted by a critical point. The algorithm combines techniques of single histogram reweighting and linear filtering. The ferromagnetic 2D Ising model is studied numerically as an illustration. In that case, the iterations reach a stationary regime with an invariant probability distribution function of temperature which peaked near the pseudocritical temperature of the specific heat. The sequence of generated temperatures is analyzed in terms of stochastic autoregressive model. The error of histogram reweighting can be better understood within the suggested model. The presented model yields a simple relation, connecting the variance of pseudocritical temperature and the parameter of linear filtering.