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Complex-Temperature Singularities in the $d=2$ Ising Model. III. Honeycomb Lattice

We study complex-temperature properties of the uniform and staggered susceptibilities $χ$ and $χ^{(a)}$ of the Ising model on the honeycomb lattice. From an analysis of low-temperature series expansions, we find evidence that $χ$ and $χ^{(a)}$ both have divergent singularities at the point $z=-1 \equiv z_{\ell}$ (where $z=e^{-2K}$), with exponents $γ_{\ell}'= γ_{\ell,a}'=5/2$. The critical amplitudes at this singularity are calculated. Using exact results, we extract the behaviour of the magnetisation $M$ and specific heat $C$ at complex-temperature singularities. We find that, in addition to its zero at the physical critical point, $M$ diverges at $z=-1$ with exponent $β_{\ell}=-1/4$, vanishes continuously at $z=\pm i$ with exponent $β_s=3/8$, and vanishes discontinuously elsewhere along the boundary of the complex-temperature ferromagnetic phase. $C$ diverges at $z=-1$ with exponent $α_{\ell}'=2$ and at $v=\pm i/\sqrt{3}$ (where $v = \tanh K$) with exponent $α_e=1$, and diverges logarithmically at $z=\pm i$. We find that the exponent relation $α'+2β+γ'=2$ is violated at $z=-1$; the right-hand side is 4 rather than 2. The connections of these results with complex-temperature properties of the Ising model on the triangular lattice are discussed.