Constrained tricritical Blume-Capel model in three dimensions

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.

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

“…We have used a combination of Metropolis, Wolff, and geometric steps, which significantly suppresses the magnitude of critical slowing down. Such simulations, together with other techniques such as the aforementioned simultaneous finite-size analysis, yield [39] the tricritical point as K t = 0.7133͑1͒ and D t = 2.0332͑3͒ on the simple-cubic lattice. The vacancy density v at the tricritical point is v = vt = 0.6485͑2͒ [39].…”

“…Such simulations, together with other techniques such as the aforementioned simultaneous finite-size analysis, yield [39] the tricritical point as K t = 0.7133͑1͒ and D t = 2.0332͑3͒ on the simple-cubic lattice. 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.…”

“…(4) becomes implicit, and a full-cluster simulation is realized by using a combination of Wolff and geometric cluster steps. It is known [39,42] that some tricritical singularities are strongly modified under this constraint. For instance, as already noted in Ref.…”

“…This problem was partly solved in Ref. [39] by means of the so-called geometric-cluster method [38]. This algorithm was developed on the basis of spatial symmetries, such as invariance under spatial inversion and rotation operations.…”

Berezinskii-Kosterlitz-Thouless-like percolation transitions in the two-dimensionalXYmodel

“…The Mott-to-superfluid transition in the Bose-Hubbard model can be characterized by the winding number of the world lines of the particles [16]. The geometric percolation [17] process has been employed to study percolation on critical substrates, such as the Ising model [18][19][20][21], the Potts model [22][23][24], the O(n) model [25], and even quantum Hall systems [26].…”