Thermodynamically self-consistent theory for the Blume-Capel model

Grollau, Sylvain; Kierlik, Édouard; Ml, Rosinberg; Tarjus, Gilles

doi:10.1103/physreve.63.041111

Cited by 30 publications

(32 citation statements)

References 37 publications

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“…II, we located [36] the tricritical point as K t = 0.7133͑1͒ and D t = 2.0313͑4͒; the expectation value of the tricritical vacancy density is vt = 0.6485͑2͒, rather close to the mean-field value 2 / 3. Consistency between these results and existing determinations [37,38], K t = 0.706͑3͒ , D t = 2.01͑1͒, and vt = 0.655͑6͒, exists within a margin of about twice the quoted errors. Here, we have applied other techniques, including a simultaneous analysis of various quantities for different systems such that parameters in common appear only once [16]; the details of these numerical analyses will be presented elsewhere [36].…”

Section: A Unconstrained Bc Modelssupporting

confidence: 85%

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Constrained tricritical Blume-Capel model in three dimensions

Deng

Blöte

2004

Phys. Rev. E

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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.

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Section: A Unconstrained Bc Modelssupporting

confidence: 85%

“…The tricritical BC model (1) has been investigated on several three-dimensional lattices, and various techniques have been developed, including the self-consistent OrnsteinZernike approximation [37] and Monte Carlo simulations [36,38].…”

Section: A Unconstrained Bc Modelsmentioning

confidence: 99%

Constrained tricritical Blume-Capel model in three dimensions

Deng

Blöte

2004

Phys. Rev. E

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“…The vacancy density v at the tricritical point is v = vt = 0.6485͑2͒ [39]. These results are consistent with estimations [40,41] from other sources K t = 0.706͑4͒, D t = 2.12͑6͒, and vt = 0.652͑6͒, within two times the error margins as quoted between parentheses.…”

Section: Monte Carlo Methods and Sampled Quantitiessupporting

confidence: 90%

Red-bond exponents of the critical and the tricritical Ising model in three dimensions

Deng

Blöte

2004

Phys. Rev. E

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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−exp͑−2K͒, where K is the Ising spin-spin coupling. Along the critical line K = K c , 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 p c =1−exp͑−2K c ͒. Further, we determine the corresponding redbond exponents as y r = 0.757͑2͒ and 0.501(5) for the critical Ising and the tricritical Blume-Capel models, respectively. On this basis, we conjecture y r =1/2 for the latter model.

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“…Figure 1 shows the phase diagram, in the -T plane, of the Blume-Capel model obtained by integration of HRT equations ͑temperatures are measured in unit of 6J/k B ). For comparison, results from SCOZA of Grollau et al ͓14,23͔ are also shown. The line of critical points ͑Néel line͒ ends at a tricritical point localized at T t ϭ0.230Ϯ0.002, c ϳ0.350Ϯ0.005.…”

Section: Resultsmentioning

confidence: 99%

Hierarchical reference theory study of the lattice restricted primitive model

2002

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A three dimensional model of point charges, named lattice restricted primitive model (LRPM), is investigated by using the hierachical reference theory of fluids. This approach, which generalizes the momentum renormalization group technique, is shown to capture the physics of the model and provides a quantitative description of the phase diagram. The comparison with recent numerical simulations and with other theoretical approaches is discussed both for the LRPM and for the Blume-Capel model, which can be seen as a screened version of LRPM. The nonuniversal crossover region close to the tricritical point is also discussed.

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Thermodynamically self-consistent theory for the Blume-Capel model

Abstract: We use a self-consistent Ornstein-Zernike approximation to study the Blume-

Cited by 30 publications

References 37 publications

Constrained tricritical Blume-Capel model in three dimensions

Constrained tricritical Blume-Capel model in three dimensions

Red-bond exponents of the critical and the tricritical Ising model in three dimensions

Hierarchical reference theory study of the lattice restricted primitive model

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