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Physical review. B, Condensed matter

Kinetic behavior of the Baxter-Wu model with quenched impurities

M. A. Novotny, David Paul Landau

Published Nov 1, 1985• 13 citations
Publisher

Abstract

We have used a standard single-spin-flip Monte Carlo method to study the kinetic behavior of the Baxter-Wu model with temperatures T>${T}_{c}$ for both zero impurities and 11.1 at. % quenched, random nonmagnetic site impurities. For the pure model we find that the linear and nonlinear relaxation are described by a dynamic exponent z=\ensuremath{\Delta}/\ensuremath{\nu}=2.0. With the addition of impurities the exponent for the linear relaxation increases to a value of z\ensuremath{\approxeq}2.8 in approximate agreement with the predictions of renormalization-group theory. The nonlinear relaxation in the impure lattice diverges more slowly than the linear relaxation but in a manner which is not completely consistent with proposed scaling theory.

DOI
10.1103/physrevb.32.5874

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We have used a standard single-spin-flip Monte Carlo method to study the kinetic behavior of the Baxter-Wu model with temperatures T>Tc{T}_{c}Tc​ for both zero impurities and 11.1 at. % quenched, random nonmagnetic site impurities. For the pure model we find that the linear and nonlinear relaxation are described by a dynamic exponent z=\ensuremath{\Delta}/\ensuremath{\nu}=2.0. With the addition of impurities the exponent for the linear relaxation increases to a value of z\ensuremath{\approxeq}2.8 in approximate agreement with the predictions of renormalization-group theory. The nonlinear relaxation in the impure lattice diverges more slowly than the linear relaxation but in a manner which is not completely consistent with proposed scaling theory.
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.
We have used an importance-sampling Monte Carlo technique to study the Baxter-Wu model with various fractions of the lattice sites occupied by random, quenched, nonmagnetic, site impurities. We found the system had long-time fluctuations which were caused by the creation and motion of domain boundaries. For this reason our study required the use of a large number of Monte Carlo steps per spin. The data were analyzed using finite-size scaling to extract the infinite-lattice critical exponents and critical amplitudes from the Monte Carlo simulation of finite lattices. This analysis of the data for the pure Baxter-Wu model yielded results which agreed well with exact results and series-expansion predictions. The addition of quenched site impurities caused a dramatic change in the critical behavior. The pure-lattice critical exponents ($\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=\frac{2}{3}$, ${\ensuremath{\gamma}}_{m}\ensuremath{\simeq}1.17$) change upon the addition of only a few percent impurities to $\ensuremath{\nu}=1.00\ifmmode\pm\else\textpm\fi{}0.07$, $\ensuremath{\alpha}\ensuremath{\lesssim}0.0$, and ${\ensuremath{\gamma}}_{m}=1.95\ifmmode\pm\else\textpm\fi{}0.08$. The phase diagram as a function of impurity concentration and an estimate for the infinite-lattice percolation limit are also given.
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.
The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.
The thermal and magnetic properties of the simple cubic Ising model with random, quenched, site impurities have been determined by Monte Carlo calculations. The transition appears to remain sharp, and the critical temperature decreases monotonically towards zero as the impurity concentration is increased towards the percolation limit. No clear evidence is found for any change in the critical exponents away from the pure lattice values. These results are discussed in terms of Harris's predictions for the critical behavior of impure systems.