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

Abstract

The low-density linked-cluster expansion of the free energy of the ferromagnetic Ising model with single-ion anisotropy, i.e., the $\ensuremath{\Delta} \ensuremath{\Sigma}{i}^{}S_{\mathrm{zi}}^{}{}_{}{}^{2}$ term, is given by $\ensuremath{-}\ensuremath{\beta}f(\ensuremath{\beta}J,\ensuremath{\beta}\ensuremath{\Delta},\ensuremath{\beta}h)=\frac{1}{2}qJ+h\ensuremath{-}\ensuremath{\Delta}+\ensuremath{\Sigma}{l=1}^{\ensuremath{\infty}}({u}^{12}\ensuremath{\mu})^{l}L_{l}(u,\ensuremath{\eta})$, with $u=\mathrm{exp}(\ensuremath{-}\frac{J}{{k}_{B}T})$, $\ensuremath{\mu}=\mathrm{exp}(\ensuremath{-}\frac{2mh}{{k}_{B}T})$, and $\ensuremath{\eta}=\mathrm{exp}(\frac{\ensuremath{\Delta}}{{k}_{B}T})$. 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 ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}}_{l}={u}^{12}^{l}L_{l}$ as a function of $\ensuremath{\Delta}$, 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 $\ensuremath{\beta}J$ with $\ensuremath{\beta}\ensuremath{\Delta}$ which estimates a tricritical value of $\ensuremath{\Delta}$. The asymptotic behavior of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}}_{l}$ is studied and from that the critical exponent $\ensuremath{\delta}$ of the $M\ensuremath{-}h$ isotherm as $h\ensuremath{\rightarrow}0$ is found. It is observed that the anisotropy determines the transition temperature, but the exponent $\ensuremath{\delta}$ is independent of the same as expected from the universality hypothesis.