Tricritical behavior of the Blume-Capel model

The thermodynamic behavior of the fcc Blume-Capel ferromagnet, $\mathcal{H}=-J\Sigma \stackrel{}{〈12〉}{S}^z\mathrm{}\left(1\right)\mathrm{}{S}^z\mathrm{}\left(2\right)\mathrm{}+\Delta \Sigma \stackrel{}{1}{\left[S^z\mathrm{}\right(1\left)\right]}^2-h\Sigma \stackrel{}{1}{S}^z\mathrm{}\left(1\right)\mathrm{}, S^z=0, \pm 1,$ 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 $\frac{k_B{T}_t}{12J}=0.\mathrm{}2615\pm 0.0070$, $\frac{{\Delta }_t}{12J}=0.\mathrm{}4716\pm 0.0010$. Tricritical exponents are consistent with ${\gamma }_t={\gamma }_t^{\prime }=1\mathrm{}$, ${\beta }_t=\frac{1}{4}$, ${\varphi }_t={\nu }_t={\alpha }_t={\alpha }_t^{\prime }=\frac{1}{2}$, in good agreement with tricritical mean-field theory and the Gaussian tricritical fixed point of Riedel and Wegner.