Complex - temperature singularities in the d = 2 Ising model : triangular and honeycomb lattices

We study complex-temperature singularities of the Ising model on the triangular and honeycomb lattices. We first discuss the complex-T phases and their boundaries. From exact results, we determine the complex-T singularities in the specific heat and magnetization. For the triangular lattice we discuss the implications of the divergence of the magnetization at the point u = − 3 (where u = z2 = e−4K ) and extend a previous study by Guttmann of the susceptibility at this point with the use of differential approximants. For the honeycomb lattice, from an analysis of low-temperature series expansions, we have found evidence that the uniform and staggered susceptibilities χ̄ and χ̄ (a) both have divergent singularities at z = −1 ≡ z`, and our numerical values for the exponents are consistent with the hypothesis that the exact values are γ ′ ` = γ ′ `,a = 2 . The critical amplitudes at this singularity were calculated. Using our exact results for α′ and β together with numerical values for γ ′ from series analyses, we find that the exponent relation α′ + 2β + γ ′ = 2 is violated at z = −1 on the honeycomb lattice; the right-hand side is consistent with being equal to 4 rather than 2. The connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed.