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Physical review. B, Solid state

Another Droplet Model for the Tricritical Point in the Blume-Capel Magnet

Dietrich Stauffer

Published Sep 1, 1972• 3 citations
Publisher

Abstract

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 ${T}_{c}$ [i.e., $A$ in $\ensuremath{\chi}=A{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$] vanishes if ${T}_{c}$ 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 $\mathcal{r}$ 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 ${T}_{c}$ 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 (${H}_{0}, {T}_{0}$) at which this first-order transition ends. The slope of ${T}_{0}(\mathcal{r})$ is finite at the tricritical point; the slope of ${H}_{0}(\mathcal{r})$ 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.

DOI
10.1103/physrevb.6.1839

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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{T}_{c}Tc​ [i.e., AAA in \ensuremathχ=A(T\ensuremath−Tc)\ensuremath−\ensuremathγ\ensuremath{\chi}=A{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}\ensuremathχ=A(T\ensuremath−Tc​)\ensuremath−\ensuremathγ] vanishes if Tc{T}_{c}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 r\ensuremath−b(Tc\ensuremath−T){\mathcal{r}}^{\ensuremath{-}b}({T}_{c}\ensuremath{-}T)r\ensuremath−b(Tc​\ensuremath−T), where r\mathcal{r}r is the distance from the tricritical point in the zero-field plane and b=(1\ensuremath−\ensuremathα)(2\ensuremath−\ensuremathα)\ensuremath≈0.5b=\frac{(1\ensuremath{-}\ensuremath{\alpha})}{(2\ensuremath{-}\ensuremath{\alpha})}\ensuremath{\approx}0.5b=(2\ensuremath−\ensuremathα)(1\ensuremath−\ensuremathα)​\ensuremath≈0.5. With this assumption, the amplitude of the spontaneous magnetization near Tc{T}_{c}Tc​ diverges for r\ensuremath→0\mathcal{r}\ensuremath{\rightarrow}0r\ensuremath→0, whereas the amplitude for the susceptibility vanishes and the amplitude for the critical isotherm remains constant; for example, the spontaneous magnetization is B(Tc\ensuremath−T)\ensuremathβB{({T}_{c}\ensuremath{-}T)}^{\ensuremath{\beta}}B(Tc​\ensuremath−T)\ensuremathβ with the amplitude B\ensuremath∝r\ensuremath−\ensuremathβbB\ensuremath{\propto}{\mathcal{r}}^{\ensuremath{-}\ensuremath{\beta}b}B\ensuremath∝r\ensuremath−\ensuremathβb. 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 r\ensuremath−b(Tc\ensuremath−T+const\ifmmode×\else\texttimes\fiM\ensuremathδ+1){\mathcal{r}}^{\ensuremath{-}b}({T}_{c}\ensuremath{-}T+\mathrm{const}\ifmmode\times\else\texttimes\fi{}{M}^{\ensuremath{\delta}+1})r\ensuremath−b(Tc​\ensuremath−T+const\ifmmode×\else\texttimes\fiM\ensuremathδ+1). This correction does not influence the second-order transition for H=0H=0H=0, but it gives the first-order transition surface for finite fields HHH and the critical line (H0,T0{H}_{0}, {T}_{0}H0​,T0​) at which this first-order transition ends. The slope of T0(r){T}_{0}(\mathcal{r})T0​(r) is finite at the tricritical point; the slope of H0(r){H}_{0}(\mathcal{r})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.
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 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.
We use a self-consistent Ornstein-Zernike approximation to study the Blume-Capel ferromagnet on three-dimensional lattices. The correlation functions and the thermodynamics are obtained from the solution of two coupled partial differential equations. The theory provides a comprehensive and accurate description of the phase diagram in all regions, including the wing boundaries in a nonzero magnetic field. In particular, the coordinates of the tricritical point are in very good agreement with the best estimates from simulation or series expansion. Numerical and analytical analysis strongly suggest that the theory predicts a universal Ising-like critical behavior along the lambda line and the wing critical lines, and a tricritical behavior governed by mean-field exponents.
We use Grassmann algebra to study the phase transition in the two-dimensional ferromagnetic Blume-Capel model from a fermionic point of view. This model presents a phase diagram with a second order critical line which becomes first order through a tricritical point, and was used to model the phase transition in specific magnetic materials and liquid mixtures of He3^33-He4^44. In particular, we are able to map the spin-1 system of the BC model onto an effective fermionic action from which we obtain the exact mass of the theory, the condition of vanishing mass defines the critical line. This effective action is actually an extension of the free fermion Ising action with an additional quartic interaction term. The effect of this term is merely to render the excitation spectrum of the fermions unstable at the tricritical point. The results are compared with recent numerical Monte-Carlo simulations.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
Abstract Since the landmark work of Lee and Yang, locating the zeros of the partition function in the complex magnetic-field plane has become a powerful method for studying phase transitions. Fisher later extended this approach to complex temperatures, and subsequent generalizations introduced other control parameters, such as the crystal field. In previous works (Moueddene et al 2024 J. Stat. Mech. 023206; 2024 Phys. Rev. E 110 064144) we applied this framework to the two- and three-dimensional Blume–Capel model–a system with a rich phase structure where a second-order critical line meets a first-order line at a tricritical point. We showed that the scaling of Lee-Yang, Fisher, and crystal-field zeros yields accurate critical exponents even for modest lattice sizes. In the present study, we extend this analysis and demonstrate that simulations need not be performed exactly at the nominal transition point to obtain reliable exponent estimates. Strikingly, small system sizes are sufficient, which not only improves methodological efficiency but also advances the broader goal of reducing the carbon footprint of large-scale computational studies.