Complex-Temperature Singularities in the $d=2$ Ising Model. II. Triangular Lattice

We investigate complex-temperature singularities in the Ising model on the triangular lattice. Extending an earlier analysis of the low-temperature series expansions for the (zero-field) susceptibility $\bar{\chi }$ by Guttmann \cite{g75} to include the use of differential approximants, we obtain further evidence in support of his conclusion that the exponent describing the divergence in $\chi$ at $u=u_e=-1\mathrm{/}3$ (where $u=e^{-4K}$) is ${\gamma }_e^{\mathrm{\prime }}=5\mathrm{/}4$ and refine his estimate of the critical amplitude. We discuss the remarkable nature of this singularity, at which the spontaneous magnetisation diverges (with exponent ${\beta }_e=-1\mathrm{/}8$) and show that it lies at the endpoint of a singular line segment constituting part of the natural boundaries of the free energy in the complex $u$ plane. Using exact results, we find that the specific heat has a divergent singularity at $u=-1\mathrm{/}3$ with exponent ${\alpha }_e^{\mathrm{\prime }}=1$, so that the relation ${\alpha }_e^{\mathrm{\prime }}+2{\beta }_e+{\gamma }_e^{\mathrm{\prime }}=2$ is satisfied. We also study the singularity at $u=u_s=-1$, where $M$ vanishes (with ${\beta }_s=3\mathrm{/}8$) and $C$ diverges logarithmically (with ${\alpha }_s^{\mathrm{\prime }}={\alpha }_s=0$).