Triangular antiferromagnetic Ising model

We solve the Ising problem on a triangular lattice with anisotropic interactions. Special consideration is given to the antiferromagnetic case. It is found that no phase transition exists if $J_1=J_2=J_3<0\mathrm{}$. Allowing a slightly different value of one of the coupling constants $J_3$, we find $k{T}_c\simeq \frac{2\left(\right|\mathrm{}{J}_1\mathrm{}|-|\mathrm{}{J}_3\mathrm{}\left|\right)\mathrm{}}{\ln 2} \mathrm{if} \left|J_3\mathrm{}\right|-\left|\mathrm{}{J}_1\right|\to 0^{-}$, while no phase transition exists if $\left|J_3\mathrm{}\right|>\mathrm{}\left|\mathrm{}{J}_1\right|$. Physical arguments to explain this behavior are also presented.