Tricritical behavior of the Blume-Capel model

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.