Critical behavior of spin-one three-dimensional Ising model with single-ion anisotropy

The low-density linked-cluster expansion of the free energy of the ferromagnetic Ising model with single-ion anisotropy, i.e., the $\Delta \Sigma i^{}S_{\mathrm{zi}}^{}{}_{}{}^2$ term, is given by $-\beta f(\beta J,\beta \Delta ,\beta h)=\frac{1}{2}qJ+h-\Delta +\Sigma {l=1\mathrm{}}^{\infty }\left(u^{12}\mu \right){}_{}{}^l{}_{}{}^{}L_l^{}{}_{}{}^{}(u,\eta )$, with $u=\exp (-\frac{J}{k_{B}T})$, $\mu =\exp (-\frac{2mh}{k_{B}T})$, and $\eta =\exp \left(\frac{\Delta }{k_{B}T}\right)$. This is a low-temperature expansion with each spin $S_i$ having magnitude one. The sixth-order series available in the literature is analyzed by evaluating the zeros of ${\tilde{L}}_l=u^{12}{}_{}{}^l{}_{}{}^{}L_l^{}{}_{}{}^{}$ as a function of $\Delta$, and from that the critical temperature is deduced. The knowledge of the critical temperature as a function of the anisotropy leads to the second-order part of the phase boundary of $\beta J$ with $\beta \Delta$ which estimates a tricritical value of $\Delta$. The asymptotic behavior of ${\tilde{L}}_l$ is studied and from that the critical exponent $\delta$ of the $M-h$ isotherm as $h\to 0$ is found. It is observed that the anisotropy determines the transition temperature, but the exponent $\delta$ is independent of the same as expected from the universality hypothesis.