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Journal of Physics A Mathematical and General

A Monte Carlo renormalisation group calculation of the dynamical exponent z for the Baxter-Wu model

Sheldon L. Katz

Published Dec 21, 1983• 0 citations
Publisher

Abstract

A Monte Carlo renormalisation group technique is used to calculate time correlation functions for the Baxter-Wu model. Initial lattices of size 24*24 and 48*48 are used. The value obtained for the dynamical exponent z is in good agreement with that obtained by means of standard Monte Carlo methods.

DOI
10.1088/0305-4470/16/18/009

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A Monte Carlo renormalisation group technique is used to calculate time correlation functions for the Baxter-Wu model. Initial lattices of size 2424 and 4848 are used. The value obtained for the dynamical exponent z is in good agreement with that obtained by means of standard Monte Carlo methods.
The effectiveness of a Monte Carlo renormalization-group method is studied by applying it to the Baxter-Wu model (Ising spins on a triangular lattice with three-spin interactions). The calculations yield three relevant eigenvalues in good agreement with exact or conjectured results. We demonstrate that the method is capable of distinguishing between models expected to be in the same universality class, when one of them (four-state Potts) exhibits logarithmic corrections to the usual power-law singularities and the other (Baxter-Wu) does not.
The short-time behavior of the Baxter-Wu model is investigated through the relaxation of the order parameter at the critical temperature. We considered Monte Carlo simulations for this model on a triangular lattice, and we studied relaxation starting from the fourfold-degenerate ground state. Using the short-time scaling formalism we found the static critical exponents beta and nu of the model and the corresponding dynamical critical exponent z. The values of the static exponents we find agree with the exact ones. To the best of our knowledge, this is the first determination of the dynamical critical exponent of the Baxter-Wu model.
We have used a Monte Carlo renormalization-group method to study the Baxter-Wu model with random, quenched, nonmagnetic site impurities. The results clearly show that the impurities change the critical exponents from the zero-impurity exact value of \ensuremath{\nu}=(2/3) to \ensuremath{\nu}=0.97\ifmmode\pm\else\textpm\fi{}0.04. The magnetic and crossover exponents do not change measurably with the addition of the impurities. We also find that the fixed point for the model with quenched impurities is further from the starting Hamiltonian than it was for the pure model.
A recent Monte Carlo renormalization-group study of the two-dimensional kinetic Ising model is extended to obtain renormalized time correlation functions from initial lattices of 16 \ifmmode\times\else\texttimes\fi{} 16 and 32 \ifmmode\times\else\texttimes\fi{} 32 spins. Different ways to estimate the dynamical exponent zzz are discussed, with our "best" estimate of $z\ensuremath{\simeq}2.23$ being consistent with the original study.