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arXiv:2610.11068

High-resolution mapping of the phase boundary of the honeycomb-lattice Blume--Capel ferromagnet

Eluiz C. M. Filho, Cláudio José DaSilva

Published Oct 8, 2026• 0 citations
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Abstract

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 $Ω(E_1,E_2)$ and field-mixing scaling, we map the first-order transition line and locate the tricritical point at $(Δ_t,T_t)=[1.4820(2),0.4022(7)]$. 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 $Δ_0=zJ/2$, the results for honeycomb, square, and triangular lattices give $r/Δ_0$ 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 $y_t=1.795(7)$, consistent with the tricritical Ising value $9/5$, while the fourth-order cumulant scaling agrees with $y_g=4/5$. Deep in the first-order regime, the cumulant approaches the two-phase limit $U_Q\to2/3$. Multicanonical simulations up to $L=64$ yield an interface tension exponent $μ=1.265(26)$, consistent with the tricritical prediction $(d-1)/y_g=5/4$. We also map the continuous phase boundary, obtaining $T_c(Δ=0)=1.157$, and identify crossover effects from Ising to tricritical scaling near the tricritical point.

DOI
10.48550/arxiv.2610.11068

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We present a high-resolution numerical study of the spin-1 Blume-Capel ferromagnet on the honeycomb lattice (z=3z=3z=3). Using the Wang-Landau algorithm to estimate the joint density of states Ω(E1,E2)Ω(E_1,E_2)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)](Δ_t,T_t)=[1.4820(2),0.4022(7)](Δt​,Tt​)=[1.4820(2),0.4022(7)]. Independent Metropolis histogram reweighting for L=36L=36L=36 and 484848 yields a crossing temperature near 0.40390.40390.4039, consistent with the Wang-Landau extrapolation. The field-mixing parameter is r=−1.585(19)r=-1.585(19)r=−1.585(19), while the coexistence-line slope gives r=−1.587(5)r=-1.587(5)r=−1.587(5). Normalizing by the zero-temperature field Δ0=zJ/2Δ_0=zJ/2Δ0​=zJ/2, the results for honeycomb, square, and triangular lattices give r/Δ0r/Δ_0r/Δ0​ between −1.057-1.057−1.057 and −1.067-1.067−1.067, 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)y_t=1.795(7)yt​=1.795(7), consistent with the tricritical Ising value 9/59/59/5, while the fourth-order cumulant scaling agrees with yg=4/5y_g=4/5yg​=4/5. Deep in the first-order regime, the cumulant approaches the two-phase limit UQ→2/3U_Q\to2/3UQ​→2/3. Multicanonical simulations up to L=64L=64L=64 yield an interface tension exponent μ=1.265(26)μ=1.265(26)μ=1.265(26), consistent with the tricritical prediction (d−1)/yg=5/4(d-1)/y_g=5/4(d−1)/yg​=5/4. We also map the continuous phase boundary, obtaining Tc(Δ=0)=1.157T_c(Δ=0)=1.157Tc​(Δ=0)=1.157, and identify crossover effects from Ising to tricritical scaling near the tricritical point.
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The thermodynamic behavior of the fcc Blume-Capel ferromagnet, $\mathcal{H}=\ensuremath{-}J\ensuremath{\Sigma}\stackrel{}{〈12〉}{S}^{z}(1){S}^{z}(2)+\ensuremath{\Delta}\ensuremath{\Sigma}\stackrel{}{1}{[{S}^{z}(1)]}^{2}\ensuremath{-}h\ensuremath{\Sigma}\stackrel{}{1}{S}^{z}(1), {S}^{z}=0, \ifmmode\pm\else\textpm\fi{}1,$ is studied by series-extrapolation techniques. By using both high- and low-temperature series, we are able to trace first- and second-order branches of the phase boundary and examine behavior at temperatures both above and below the phase transition. We find a tricritical point at $\frac{{k}_{B}{T}_{t}}{12J}=0.2615\ifmmode\pm\else\textpm\fi{}0.0070$, $\frac{{\ensuremath{\Delta}}_{t}}{12J}=0.4716\ifmmode\pm\else\textpm\fi{}0.0010$. Tricritical exponents are consistent with ${\ensuremath{\gamma}}_{t}={\ensuremath{\gamma}}_{t}^{\ensuremath{'}}=1$, ${\ensuremath{\beta}}_{t}=\frac{1}{4}$, ${\ensuremath{\phi}}_{t}={\ensuremath{\nu}}_{t}={\ensuremath{\alpha}}_{t}={\ensuremath{\alpha}}_{t}^{\ensuremath{'}}=\frac{1}{2}$, in good agreement with tricritical mean-field theory and the Gaussian tricritical fixed point of Riedel and Wegner.
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.
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