The microcanonical partition function ${\ensuremath{\Omega}}_{\ensuremath{\alpha},N}(E)$ of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios $\ensuremath{\alpha}$ close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ spins with $L\ensuremath{\in}{8,10,12,14,18}$ was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields ${\ensuremath{\alpha}}^{(t)}=2.84479\ifmmode\pm\else\textpm\fi{}0.00030$ for the tricritical coupling ratio, and ${k}_{B}{T}^{(t)}/J=1.4182\ifmmode\pm\else\textpm\fi{}0.0055$ for the tricritical transition temperature. The results are compared with values obtained on fcc lattices.
An alternative to Reatto's double-droplet model fortricritical points is proposed for the Blume-Capel ferromagnet in order to get agreement with the result of Saul and Wortis, that the amplitude of the susceptibility divergence at
Tc [i.e.,
A in $\ensuremath{\chi}=A{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$] vanishes if
Tc approaches the tricritical point. In our droplet-model description of the Blume-Capel magnet we employ only one and not two types of droplets, whereas Reatto used two. Our semiphenomenological theory assumes a phase diagram as given by Saul and Wortis from series expansions. First we assume that the droplet surface tension varies as ${\mathcal{r}}^{\ensuremath{-}b}({T}_{c}\ensuremath{-}T)$, where
r is the distance from the tricritical point in the zero-field plane and $b=\frac{(1\ensuremath{-}\ensuremath{\alpha})}{(2\ensuremath{-}\ensuremath{\alpha})}\ensuremath{\approx}0.5$. With this assumption, the amplitude of the spontaneous magnetization near
Tc diverges for $\mathcal{r}\ensuremath{\rightarrow}0$, whereas the amplitude for the susceptibility vanishes and the amplitude for the critical isotherm remains constant; for example, the spontaneous magnetization is $B{({T}_{c}\ensuremath{-}T)}^{\ensuremath{\beta}}$ with the amplitude $B\ensuremath{\propto}{\mathcal{r}}^{\ensuremath{-}\ensuremath{\beta}b}$. Also for the first-order phase transition various tricritical exponents are expressed by the usual critical exponents. Then we add to the surface tension a magnetization-dependent term; for a particularly simple choice the surface tension now varies as ${\mathcal{r}}^{\ensuremath{-}b}({T}_{c}\ensuremath{-}T+\mathrm{const}\ifmmode\times\else\texttimes\fi{}{M}^{\ensuremath{\delta}+1})$. This correction does not influence the second-order transition for
H=0, but it gives the first-order transition surface for finite fields
H and the critical line (
H0,T0) at which this first-order transition ends. The slope of
T0(r) is finite at the tricritical point; the slope of
H0(r) diverges there. Our predicted exponents for the susceptibility at the second-order transition and for the magnetization at the first-order transition for zero field agree roughly with results of Saul and Wortis; nevertheless, our theory is still rather speculative.
Using a 1/z expansion we study the correlation of fluctuations in a spin-1 Ising model with anisotropy. Previously, the 1/z expansion has been applied only to the second-order phase transitions of ferromagnets and granular superconductors. By imposing conditions on the free energy F and the parameter x=1-〈
S1z2〉, we use the 1/z technique to examine both the first- and second-order parts of the Blume-Capel phase diagram. In agreement with many numerical simulations and experiments on $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures, the value of x at the tricritical point is not shifted from its mean-field value of 2/3 to order 1/z. On the other hand, the tricritical values of the anisotropy parameter A and of the transition temperature
Tc are altered by the correlation of fluctuations. At least to order 1/z, all the derivatives of
Tc are continuous across the tricritical point.
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 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.
The critical behavior of the Blume-Capel model for
d=2,3 is analyzed using a renormalization transformation based on Kadanoff's lower-bound transformation. There are fixed points associated with first-order, second-order, and tricritical phase transitions. The discontinuity fixed point onto which the line of first-order phase transition is mapped has two relevant eigenoperators with eigenvalue
2d, corresponding to the discontinuities in $〈\ensuremath{\sigma}〉$ and $〈{\ensuremath{\sigma}}^{2}〉$. Quite different values of the variational parameters in the renormalization transformation optimize the free energy at each fixed point, making it difficult to calculate crossover behavior with the present approach.