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

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 $\Omega (E_1,E_2)$ and field-mixing scaling, we map the first-order transition line and locate the tricritical point at $({\Delta }_t,T_t)=[1.4820(2),0.4022(7)]$. 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 ${\Delta }_0=zJ/2$, the results for honeycomb, square, and triangular lattices give $r/{\Delta }_0$ 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 $U_Q\to 2/3$. Multicanonical simulations up to $L=64$ yield an interface tension exponent $\mu =1.265(26)$, consistent with the tricritical prediction $(d-1)/y_g=5/4$. We also map the continuous phase boundary, obtaining $T_c(\Delta =0)=1.157$, and identify crossover effects from Ising to tricritical scaling near the tricritical point.