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

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

1994-11-14

arxiv.org/abs/hep-lat/9411023v2

J.Phys. A29 (1996) 803-823

Physics

High Energy Physics

High Energy Physics - Lattice

latex file, 25 pages of text plus figures appended to end of file. (Further references have been included to important earlier

Scientific paper

10.1088/0305-4470/29/4/009

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 $u=u_e=-1/3$ (where $u = e^{-4K}$) is $\gamma_e'=5/4$ and refine his estimate of the critical amplitude. We discuss the remarkable nature of this singularity, at which the spontaneous magnetisation diverges (with exponent $\beta_e=-1/8$) 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 $u=-1/3$ with exponent $\alpha_e'=1$, so that the relation $\alpha_e'+2\beta_e+\gamma_e'=2$ is satisfied. We also study the singularity at $u=u_s=-1$, where $M$ vanishes (with $\beta_s=3/8$) and $C$ diverges logarithmically (with $\alpha_s' = \alpha_s = 0$).

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