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Physical review. B, Condensed matter

Critical behavior of the Baxter-Wu model with quenched impurities

M. A. Novotny, David Paul Landau

Published Aug 1, 1981• 71 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.24.1468

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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 (\ensuremathν=23\ensuremath{\nu}=\frac{2}{3}\ensuremathν=32​, \ensuremathα=23\ensuremath{\alpha}=\frac{2}{3}\ensuremathα=32​, \ensuremathγm\ensuremath≃1.17{\ensuremath{\gamma}}_{m}\ensuremath{\simeq}1.17\ensuremathγm​\ensuremath≃1.17) change upon the addition of only a few percent impurities to \ensuremathν=1.00\ifmmode±\else\textpm\fi0.07\ensuremath{\nu}=1.00\ifmmode\pm\else\textpm\fi{}0.07\ensuremathν=1.00\ifmmode±\else\textpm\fi0.07, \ensuremathα\ensuremath≲0.0\ensuremath{\alpha}\ensuremath{\lesssim}0.0\ensuremathα\ensuremath≲0.0, and \ensuremathγm=1.95\ifmmode±\else\textpm\fi0.08{\ensuremath{\gamma}}_{m}=1.95\ifmmode\pm\else\textpm\fi{}0.08\ensuremathγm​=1.95\ifmmode±\else\textpm\fi0.08. The phase diagram as a function of impurity concentration and an estimate for the infinite-lattice percolation limit are also given.
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 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.
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.
We studied the pure and dilute Baxter–Wu (BW) models using the Wang–Landau (WL) sampling method to calculate the density-of-states (DOS). We first used the exact result for the DOS of the Ising model to test our code. Then we calculated the DOS of the dilute Ising model to obtain a phase diagram, in good agreement with previous studies. We calculated the energy distribution, together with its first, second and fourth moments, to give the specific heat and the energy fourth order cumulant, better known as the Binder parameter, for the pure BW model. For small samples, the energy distribution displayed a doubly peaked shape, and finite size scaling analysis showed the expected reciprocal scaling of the positions of the peaks with L . The energy distribution yielded the expected BW α = 2/3 critical exponent for the specific heat. The Binder parameter minimum appeared to scale with lattice size L with an exponent θ B equal to the specific heat exponent. Its location (temperature) showed a large correction-to-scaling term θ 1 = 0.248 ± 0.025. For the dilute BW model we found a clear crossover to a single peak in the energy distribution even for small sizes and the expected α = 0 was recovered.
The influence of quenched nonmagnetic impurities on phase transitions in the three-dimensional Potts model with the number of spin states q = 3 is investigated using the Wolff single-cluster algorithm of the Monte Carlo method. The systems with linear sizes L = 20–44 at the spin concentrations p = 1.0, 0.9, 0.8, and 0.7 are analyzed. It is demonstrated with the use of the method of fourth-order Binder cumulants that the second-order phase transition occurs in the model under consideration at the spin concentrations p = 0.9, 0.8, and 0.7 and that the first-order phase transition is observed in the pure model ( p = 1.0). The static critical exponents α (heat capacity), γ (susceptibility), β (magnetization), and ν (correlation length) are calculated in the framework of the finite-size scaling theory. The problem regarding the universality classes of the critical behavior of weakly diluted systems is discussed.
We report single-cluster Monte Carlo simulations of the Ising model on three-dimensional Poissonian random lattices with up to $128000\ensuremath{\approx}{50}^{3}$ sites which are linked together according to the Voronoi-Delaunay prescription. For each lattice size quenched averages are performed over 96 realizations. By using reweighting techniques and finite-size scaling analyses we investigate the critical properties of the model in the close vicinity of the phase transition point. Our random lattice data provide strong evidence that, for the available system sizes, the resulting effective critical exponents are indistinguishable from recent high-precision estimates obtained in Monte Carlo studies of the Ising model and ${\ensuremath{\varphi}}^{4}$ field theory on three-dimensional regular cubic lattices.