On the magnetization of the triangular lattice Ising model with triplet interactions

The analytic properties of the spontaneous magnetization $M$ for the triangular lattice Ising model with pure triplet interactions are investigated. In particular, it is shown that $M$ is a solution of Legendre's elliptic modular equation of degree 3. The following explicit hypergeometric formula for $M$ is also derived: $M$ = $M$($k$$_{1}^{2}$) = (1-$k$$_{1}^{2}$)$^{\frac{3}{8}}$ [1-($_{4}^{3}$k$_{1}^{2})$_{2}$F$$_{1}$($\frac{1}{2}$, $\frac{1}{6}$; $\frac{4}{3}$; $k$_{1}^{2})$^{2}$]$^{-\frac{1}{2}}$, where $k$$_{1}^{2}$ = 16$u$(1 - $u$)$^{2}$/(1 + $u$)$^{4}$, and $u$ = exp(-4$J$/$k$_{\text{B}}$T$). From these basic results it is proved that $M$($k$$_{1}^{2})$ and $M(u)$ satisfy linear Fuchsian differential equations of fourth order. The detailed behaviour of $M$ in the neighbourhood of its physical and non-physical singularities is also studied. It is found that the behaviour of $M$ near the critical point $u$$_{\text{c}}$ = 3-2$\surd $2 is described by an analytic continuation of the type $M$ = $\epsilon ^{\frac{1}{12}}$[$f$$_{0}$($\epsilon)$ + $\epsilon ^{\frac{2}{3}}$f$_{1}$($\epsilon $) + $\epsilon ^{\frac{4}{3}}$f$_{2}$($\epsilon $)], as $\epsilon $ $\rightarrow $ 0+, where $\epsilon $ = 1 - ($u/u$$_{\text{c}}$), and the functions $f_{i}$$(\epsilon)$ $(i=0,1,2)$ are analytic at $\epsilon $ = 0. Finally, it is noted that the spontaneous magnetization for the square lattice Ising model with pair interactions is a solution of Landen's modular equation of degree 2.