We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
We present a high-resolution numerical study of the spin-1 Blume-Capel ferromagnet on the honeycomb lattice (
z=3). Using the Wang-Landau algorithm to estimate the joint density of states
Ω(E1,E2) and field-mixing scaling, we map the first-order transition line and locate the tricritical point at
(Δt,Tt)=[1.4820(2),0.4022(7)]. Independent Metropolis histogram reweighting for
L=36 and
48 yields a crossing temperature near
0.4039, consistent with the Wang-Landau extrapolation. The field-mixing parameter is
r=−1.585(19), while the coexistence-line slope gives
r=−1.587(5). Normalizing by the zero-temperature field
Δ0=zJ/2, the results for honeycomb, square, and triangular lattices give
r/Δ0 between
−1.057 and
−1.067, about 6% larger in magnitude than the mean-field value. Simple-cubic estimates are compatible with this range but do not resolve dimensionality dependence. Finite-size scaling yields
yt=1.795(7), consistent with the tricritical Ising value
9/5, while the fourth-order cumulant scaling agrees with
yg=4/5. Deep in the first-order regime, the cumulant approaches the two-phase limit
UQ→2/3. Multicanonical simulations up to
L=64 yield an interface tension exponent
μ=1.265(26), consistent with the tricritical prediction
(d−1)/yg=5/4. We also map the continuous phase boundary, obtaining
Tc(Δ=0)=1.157, and identify crossover effects from Ising to tricritical scaling near the tricritical point.
The phase diagram of the two-dimensional Blume-Capel model is investigated by using the technique of phenomenological finite-size scaling. The location of the tricritical point and the values of the critical and tricritical exponents are determined. The location of the tricritical point (
Tt=0.610\ifmmode\pm\else\textpm\fi{}0.005,
Dt=1.9655\ifmmode\pm\else\textpm\fi{}0.0010) is well outside the error bars for the value quoted in previous Monte Carlo simulations but in excellent agreement with more recent Monte Carlo renormalization-group results. The values of the critical and tricritical exponents, with the exception of the leading thermal tricritical exponent, are in excellent agreement with previous calculations, conjectured values, and Monte Carlo renormalization-group studies.
This article investigates the pseudo transitions of the Blume-Capel model on two-dimensional finite-size lattices. By employing the Wang-Landau sampling method and microcanonical inflection point analysis, we identified the positions of phase transitions as well as higher-order phase transitions. Through Metropolis sampling and canonical ensemble analysis, we determined the geometric characteristics of the system at these transition points. When the crystal field parameter
D exceeds 1.965, crossing the tricritical point, no third-order dependent phase transition is observed. However, a fourth-order independent transition was identified in the high-temperature region, and through Metropolis sampling analysis, a phase transition from the ordered paramagnetic phase to the disordered paramagnetic phase was confirmed, enhancing the phase diagram. Furthermore, the positions of the third-order phase transition obtained from both microcanonical and canonical analyses are consistent and mutually corroborative. We speculate that third-order dependent transitions vanish in the presence of strong first-order phase transitions.
The effects of bond randomness on the phase diagram and critical behavior of the square lattice ferromagnetic Blume-Capel model are discussed. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method for many values of the crystal field, restricted here in the second-order phase-transition regime of the pure model. For the random-bond version several disorder strengths are considered. We present phase diagram points of both pure and random versions and for a particular disorder strength we locate the emergence of the enhancement of ferromagnetic order observed in an earlier study in the ex-first-order regime. The critical properties of the pure model are contrasted and compared to those of the random model. Accepting, for the weak random version, the assumption of the double-logarithmic scenario for the specific heat we attempt to estimate the range of universality between the pure and random-bond models. The behavior of the strong disorder regime is also discussed and a rather complex and yet not fully understood behavior is observed. It is pointed out that this complexity is related to the ground-state structure of the random-bond version.
Abstract We investigate the density of states (DOS) in an antiferromagnetic spin‐system on a square lattice described by the Blume–Capel (BC) model. We use a new and very efficient simulation method, proposed by Wang and Landau, in which we estimate very precisely DOS by sampling in the space of energy. Then we calculate the thermodynamical averages like internal energy, free energy, specific heat and entropy. The BC model exhibits multicritical behaviour such as first‐ or second‐order transitions and tricritical points. It is known that the ground state of the model can exhibit two kinds of staggered antiferromagnetic phases: AF 1 (two interpenetrating lattices with S = –1 and S = 1) and AF 2 ( S = –1 and S = 0 for H < 0; S = 1 and S = 0 for H > 0). We analyze the coexistence of such phases at finite temperatures and determine border lines between them. To understand the microscopic nature of such boundaries we present also some results obtained with the standard Monte Carlo method. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)