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 of the Ising model on the honeycomb lattice. From an analysis of low-temperature series expansions, we find evidence that χ and χ both have divergent singularities at the point z = −1 ≡ zl (where z = e), with exponents γ l = γ l,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 βl = −1/4, vanishes continuously at z = ±i with exponent βs = 3/8, and vanishes discontinuously elsewhere along the boundary of the complex-temperature ferromagnetic phase. C diverges at z = −1 with exponent α l = 2 and at v = ±i/ √ 3 (where v = tanhK) with exponent αe = 1, and diverges logarithmically at z = ±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. ∗email: [email protected] ∗∗email: [email protected]