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

We present a high-resolution numerical study of the spin-1 Blume–Capel ferromagnet on the honeycomb lattice ($z=3$). Wang–Landau estimates of the joint density of states $\mathrm{\Omega }(E_1,E_2)$, combined with field mixing, are used to map the first-order transition line and locate the tricritical point at $({\mathrm{\Delta }}_t,T_t)=[1.4819(2),0.4022(7)]$. Histogram reweighting of Metropolis simulations for $L=36$ and $48$ provides an independent consistency check, with a crossing temperature near $0.4039$, compatible with the Wang–Landau extrapolation. From the field-mixing analysis we obtain $r=-1.586(19)$, while the slope of the multicanonical coexistence line yields the consistent and more precise estimate $r=-1.587(5)$. Normalizing by the zero-temperature field ${\mathrm{\Delta }}_0=zJ\mathrm{/}2$, the available honeycomb, square, triangular, and simple-cubic estimates cluster near $r\mathrm{/}{\mathrm{\Delta }}_0\simeq -1.06$, corresponding to a magnitude about $6\mathrm{\%}$ larger than the mean-field value $-1$; no systematic dependence on coordination number or dimensionality is resolved over the range examined. Finite-size scaling of ${\sigma }_Q$ gives $y_t=1.795(7)$, consistent with the tricritical Ising value $9\mathrm{/}5$, while the fourth-order cumulant scaling is compatible with $y_g=4\mathrm{/}5$. Deep in the first-order regime, the cumulant approaches the expected two-phase limit $U_Q\to 2\mathrm{/}3$. Multicanonical simulations up to $L=64$, using weights constructed directly from the Wang–Landau density of states for $L\le 24$ and rescaled and refined for larger systems, yield an interface-tension exponent $\mu =1.265(26)$, consistent with the tricritical prediction $(d-1)\mathrm{/}{y}_g=5\mathrm{/}4$. Finally, we map the continuous branch of the phase boundary from the same density-of-states data, obtaining $T_c(\mathrm{\Delta }=0)=1.157$; estimates sufficiently close to the tricritical point are affected by crossover from Ising to tricritical scaling.