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

TL;DR

This paper investigates complex-temperature singularities of the Ising model on the honeycomb lattice, revealing divergent behaviors and critical exponents at specific complex points, and compares these findings with other lattice models.

Contribution

It provides new insights into the complex-temperature singularities and critical exponents of the honeycomb lattice Ising model, including exact and series-based analyses.

Findings

Divergent singularities at z=-1 with exponent 5/2

Magnetisation diverges at z=-1 with exponent -1/4

Specific heat diverges at z=-1 with exponent 2

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 $\gamma_{\ell}'= \gamma_{\ell,a}'=5/2$. 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 $\beta_{\ell}=-1/4$, 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 $\alpha_{\ell}'=2$ 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 $\alpha'+2\beta+\gamma'=2$ 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.