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.
We study the S=1 Ising model, equivalent to the three-state lattice-gas model, with nearest-neighbor, pairwise interactions on a two-dimensional, triangular lattice. We pay particular attention to the antiferromagnetic phase diagrams. We show its relation to other well-studied models (S=(1/2 Ising, Blume-Capel, Blume-Emery-Griffiths), classify the ground states, and calculate finite-temperature phase diagrams using transfer matrices and finite-size scaling for infinite strips of three and six sites width. The phase diagrams are quite complicated, with surfaces of first- and second-order transitions that intersect along lines of multicritical points of various kinds, providing a rich laboratory for studying a number of first-order phase transitions, critical and multicritical phenomena within the framework of one single model.
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.
We investigate the tricritical scaling behavior of the two-dimensional spin-1 Blume-Capel model by using the Wang-Landau method of measuring the joint density of states for lattice sizes up to 48×48 sites. We find that the specific heat deep in the first-order area of the phase diagram exhibits a double-peak structure of the Schottky-like anomaly appearing with the transition peak. The first-order transition curve is systematically determined by employing the method of field mixing in conjunction with finite-size scaling, showing a significant deviation from the previous data points. At the tricritical point, we characterize the tricritical exponents through finite-size-scaling analysis including the phenomenological finite-size scaling with thermodynamic variables. Our estimation of the tricritical eigenvalue exponents, yt=1.804(5), yg=0.80(1), and yh=1.925(3), provides the first Wang-Landau verification of the conjectured exact values, demonstrating the effectiveness of the density-of-states-based approach in finite-size scaling study of multicritical phenomena.