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

Abstract

We study complex-temperature properties of the uniform and staggered susceptibilities $\chi$ and ${\chi }^{(a)}$ of the Ising model on the honeycomb lattice. From an analysis of low-temperature series expansions, we find evidence that $\chi$ and ${\chi }^{(a)}$ both have divergent singularities at the point $z=-1\equiv z_{\ell }$ (where $z=e^{-2K}$), with exponents ${\gamma }_{\ell }^{\prime }={\gamma }_{\ell ,a}^{\prime }=5/2$. 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 ${\beta }_{\ell }=-1/4$, vanishes continuously at $z=\pm i$ with exponent ${\beta }_s=3/8$, and vanishes discontinuously elsewhere along the boundary of the complex-temperature ferromagnetic phase. $C$ diverges at $z=-1$ with exponent ${\alpha }_{\ell }^{\prime }=2$ and at $v=\pm i/\sqrt{3}$ (where $v=\tanh K$) with exponent ${\alpha }_e=1$, and diverges logarithmically at $z=\pm i$. We find that the exponent relation ${\alpha }^{\prime }+2\beta +{\gamma }^{\prime }=2$ 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.

22 pages, latex, figures appended after the end of the text as a compressed, uuencoded postscript file

References in corpus (2)

• Complex-Temperature Singularities of the Susceptibility in the $d=2$ Ising Model. I. Square Lattice
• Low-Temperature Series Expansions for the Spin-1 Ising Model

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