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.
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 investigate the spin-1 Blume-Capel model on an infinite strip of the triangular lattice using the transfer-matrix method combined with a sparse-matrix factorization technique. Through finite-size scaling analysis of numerically exact spectra for strip widths up to L = 19 , we accurately locate the tricritical point improving upon recent Monte Carlo estimates. In the first-order regime, we observe exponential scaling of the spectral gap, reflecting the linear growth of interfacial tension as the temperature decreases below the tricritical point. Finally, we validate our tricritical point estimate through precise agreement with conformal field theory predictions for the tricritical Ising universality class. Our results underscore the continued utility of the transfer-matrix approach for studying phase transitions in complex lattice models.
The Blume-Caple model on the triangular lattice is considered in this paper. Using the (perturbative) real-space renormalization group (RSRG) technique we analytically obtain the running of the parameters of the model, i.e., the exchange integral J and the crystal field Δ. The observations show that the first-order calculations are insufficient to capture the tricritical phenomena, and the RG analysis based on second-order expansion of the partition function show qualitative agreement with the previous works. This reveals that the second-order perturbation works appropriately, to obtain the qualitative features of the phase diagram of the model, e.g., a tricritical point which is unstable towards the Ising critical point from one side (which forms the line of second order, separating the ferromagnetic and paramagnetic phases) and the critical point at and from the other side (which forms the line of first-order transition separating ferromagnetic and zero state phases). This analysis shows also that there is another unstable critical point at , with the RG eigenvalue in the Δ-direction (towards S = 0 zero state and paramagnetic phases).
We perform Monte-Carlo simulations of the anti-ferromagnetic (AF) spin-1 Blume- Capel (BC) model and the AF Ising model on triangular lattice. We estimate the exact critical magnetic fields for both models at zero temperature using the Wang-Landau sampling method. We also show the phase diagrams and the critical lines for the models using the joint density functions. We find that the shapes of critical lines for the models are identical, but the phase transitions across the critical lines are different.
The effects of bond randomness on the universality aspects of a two-dimensional (d = 2) Blume-Capel model embedded in the triangular lattice are discussed. The system is studied numerically in both its first- and second-order phase-transition regimes by a comprehensive finite-size scaling analysis for a particularly suitable value of the disorder strength. We find that our data for the second-order phase transition, emerging under random bonds from the second-order regime of the pure model, are compatible with the universality class of the two-dimensional (2D) random Ising model. Furthermore, we find evidence that, the second-order transition emerging under bond randomness from the first-order regime of the pure model, belongs again to the same universality class. Although the first finding reinforces the scenario of strong universality in the 2D Ising model with quenched disorder, the second is in difference from the critical behavior, emerging under randomness, in the cases of the ex-first-order transitions of the Potts model. Finally, our results verify previous renormalization-group calculations on the Blume-Capel model with disorder in the crystal-field coupling.