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Physical Review E

Constrained tricritical Blume-Capel model in three dimensions

Youjin Deng, Henk W. J. Blöte

Published Oct 21, 2004• 37 citations
Publisher

Abstract

Using the Wolff and geometric cluster Monte Carlo methods, we investigate the tricritical Blume-Capel model in three dimensions. Since these simulations conserve the number of vacancies and thus effectively introduce a constraint, we generalize the Fisher renormalization for constrained critical behavior to tricritical systems. We observe that, indeed, the tricritical behavior is significantly modified under this constraint. For instance, at tricriticality, the specific heat has only a finite cusp and the Binder ratio assumes a different value from that in unconstrained systems. Since 3 is the upper tricritical dimensionality of Ising systems, we expect that the mean-field theory correctly predicts a number of universal parameters in three dimensions. Therefore, we calculate the partition sum of the mean-field tricritical Blume-Capel model, and accordingly obtain the exact value of the Binder ratio. Under the constraint, we show that this mean-field tricritical system reduces to the mean-field critical Ising model. However, our three-dimensional data do not agree with this mean-field prediction. Instead, they are successfully explained by the generalized Fisher renormalization mechanism.

DOI
10.1103/physreve.70.046111

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Using the Wolff and geometric cluster Monte Carlo methods, we investigate the tricritical Blume-Capel model in three dimensions. Since these simulations conserve the number of vacancies and thus effectively introduce a constraint, we generalize the Fisher renormalization for constrained critical behavior to tricritical systems. We observe that, indeed, the tricritical behavior is significantly modified under this constraint. For instance, at tricriticality, the specific heat has only a finite cusp and the Binder ratio assumes a different value from that in unconstrained systems. Since 3 is the upper tricritical dimensionality of Ising systems, we expect that the mean-field theory correctly predicts a number of universal parameters in three dimensions. Therefore, we calculate the partition sum of the mean-field tricritical Blume-Capel model, and accordingly obtain the exact value of the Binder ratio. Under the constraint, we show that this mean-field tricritical system reduces to the mean-field critical Ising model. However, our three-dimensional data do not agree with this mean-field prediction. Instead, they are successfully explained by the generalized Fisher renormalization mechanism.
We investigate several tricritical models on the square lattice by means of Monte Carlo simulations. These include the Blume-Capel model, Baxter's hard-square model, and the q=1 , 3, and 4 Potts models with vacancies. We use a combination of the Wolff and geometric cluster methods, which conserves the total number of vacancies or lattice-gas particles and suppresses critical slowing down. Several quantities are sampled, such as the specific heat C and the structure factor C(s) , which accounts for the large-scale spatial inhomogeneity of the energy fluctuations. We find that the constraint strongly modifies some of the critical singularities. For instance, the specific heat C reaches a finite value at tricriticality, while C(s) remains divergent as in the unconstrained system. We are able to explain these observed constrained phenomena on the basis of the Fisher renormalization mechanism generalized to include a subleading relevant thermal scaling field. In this context, we find that, under the constraint, the leading thermal exponent y(t1) is renormalized to 2- y(t1) , while the subleading exponent y(t2) remains unchanged.
We investigate the location of the critical and tricritical points of the three-dimensional Blume-Capel model by analyzing the behavior of the first Lee-Yang zero, the density of partition function zeros, and higher-order cumulants of the magnetization. Our analysis is conducted through Monte-Carlo simulations, intentionally using only small system sizes. We demonstrate that this approach yields excellent results for studying the critical behavior of the model. Our findings indicate that, at the tricritical point, where logarithmic corrections are anticipated, the numerical results align closely with the theoretical exponents describing these corrections. These expected values are then employed to accurately determine the coordinates of the tricritical point. At the model's critical point, the corrections correspond to those of the three-dimensional Ising model criticality, which we also use to precisely ascertain the critical temperature at zero crystal field. Additionally, we utilize more traditional thermodynamic quantities to validate the self-consistency of our analysis.
The microcanonical partition function ${\ensuremath{\Omega}}_{\ensuremath{\alpha},N}(E)$ of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios $\ensuremath{\alpha}$ close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ spins with $L\ensuremath{\in}{8,10,12,14,18}$ was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields ${\ensuremath{\alpha}}^{(t)}=2.84479\ifmmode\pm\else\textpm\fi{}0.00030$ for the tricritical coupling ratio, and ${k}_{B}{T}^{(t)}/J=1.4182\ifmmode\pm\else\textpm\fi{}0.0055$ for the tricritical transition temperature. The results are compared with values obtained on fcc lattices.
Using a Monte Carlo renormalization-group (MCRG) method we study tricritical behavior in two very different two-dimensional spin models: the Blume-Capel ferromagnet and the next-nearest-neighbor Ising antiferromagnet. The MCRG method is used to locate accurately tricritical points independent of the convergence of eigenvalue estimates. Despite the different symmetries and the different renormalization transformations used for the two models, in both cases we find four relevant tricritical eigenvalues which are essentially identical for both models. For the antiferromagnet, there is no indication of any decomposition of the tricritical point as predicted by mean-field theory, for ratios of intrasublattice to intersublattice coupling as small as (1/8).
Using the Wolff and geometric cluster algorithms and finite-size scaling analysis, we investigate the critical Ising and the tricritical Blume-Capel models with nearest-neighbor interactions on the simple-cubic lattice. The sampling procedure involves the decomposition of the Ising configuration into geometric clusters, each of which consists of a set of nearest-neighboring spins of the same sign connected with bond probability p. These clusters include the well-known Kasteleyn-Fortuin clusters as a special case for p=1-exp(-2K) , where K is the Ising spin-spin coupling. Along the critical line K=Kc , the size distribution of geometric clusters is investigated as a function of p . We observe that, unlike in the case of two-dimensional tricriticality, the percolation threshold in both models lies at pc =1-exp(-2Kc) . Further, we determine the corresponding red-bond exponents as yr =0.757(2) and 0.501(5) for the critical Ising and the tricritical Blume-Capel models, respectively. On this basis, we conjecture yr =1/2 for the latter model.
Abstract We perform Monte Carlo simulations, combining both the Wang–Landau and the Metropolis algorithms, to investigate the phase diagrams of the Blume–Capel model on different types of nonregular lattices (Lieb lattice (LL), decorated triangular lattice (DTL), and decorated simple cubic lattice (DSC)). The nonregular character of the lattices induces a double transition (reentrant behavior) in the region of the phase diagram at which the nature of the phase transition changes from first-order to second-order. A physical mechanism underlying this reentrance is proposed. The large-scale Monte Carlo simulations are performed with the finite-size scaling analysis to compute the critical exponents and the critical Binder cumulant for three different values of the anisotropy Δ / J ∈ 0 , 1 , 1.34 ( for LL ) , 1.51 ( for DTL and DSC ) , showing thus no deviation from the standard Ising universality class in two and three dimensions. We report also the location of the tricritical point to considerable precision: (Δ t / J = 1.3457(1); k B T t / J = 0.309(2)), (Δ t / J = 1.5766(1); k B T t / J = 0.481(2)), and (Δ t / J = 1.5933(1); k B T t / J = 0.569(4)) for LL, DTL, and DSC, respectively.