Eight-vertex model on the honeycomb lattice

The most general vertex model defined on a honeycomb lattice is the eight-vertex model. In this paper it is shown that the symmetric eight-vertex model reduces to an Ising model with a nonzero real or pure imaginary magnetic field H. The equivalent Ising model is either ferromagnetic with e2H/kT real or antiferromagnetic with e2H/kT unimodular. The exact transition temperature and the order of phase transition in the former case are determined. As an application of the result we verify the absence of a phase transition in the monomer-dimer system on the honeycomb lattice.