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arXiv:physics/0402090

Another Monte Carlo Renormalization Group Algorithm

Donohue, John P.

Published Feb 18, 2004• 0 citations
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Abstract

A Monte Carlo Renormalization Group algorithm is used on the Ising model to derive critical exponents and the critical temperature. The algorithm is based on a minimum relative entropy iteration developed previously to derive potentials from equilibrium configurations. This previous algorithm is modified to derive useful information in an RG iteration. The method is applied in several dimensions with limited success. Required accuracy has not been achieved, but the method is interesting.

DOI
10.48550/arxiv.physics/0402090

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A Monte Carlo Renormalization Group algorithm is used on the Ising model to derive critical exponents and the critical temperature. The algorithm is based on a minimum relative entropy iteration developed previously to derive potentials from equilibrium configurations. This previous algorithm is modified to derive useful information in an RG iteration. The method is applied in several dimensions with limited success. Required accuracy has not been achieved, but the method is interesting.
A novel dynamic Monte Carlo renormalization-group method is devised. Application to the kinetic Ising model in one dimension yields a value of zzz, the dynamic critical exponent, which is in excellent agreement with the exact result. In two dimensions we find $z=2.22\ifmmode\pm\else\textpm\fi{}0.13$ with the most probable value z=2.17z=2.17z=2.17.
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.