Eight - vertex model and Ising model in a non - zero magnetic field : honeycomb lattice

The known equivalence of the honeycomb eight-vertex model with an Ising model in a non-zero magnetic field is derived via a direct mapping. Compared with a previous derivation which uses the generalised weak-graph transformation, the new method is simpler and more direct, and can be extended to other considerations. The eight-vertex model on the honeycomb lattice is a general lattice model playing the role of the 16-vertex model for the square lattice. The honeycomb problem was first considered by Wu [I], who used a generalised weak-graph transformation [2-41 to study its soluble cases. The honeycomb eight-vertex model has since proven to be a useful tool in deducing exact results for a number of physical problems. They include the obtaining of a closed-form expression for the critical frontier of the antiferromagnetic Ising model [5], the establishment of the effect of three-body interactions on the critical behaviour of the coexistence curve diameter of a lattice gas [6], the determination of the exact phase diagram of a spin system with twoand three-site interactions [7] and an exact analysis of the spin-1 Blume-Emery-Griffiths model [8]. A key step in all these studies is the use of the aforementioned equivalence of the eight-vertex model with an Ising model in a non-zero magnetic field. While it is fairly easy to deduce this equivalence for a special subspace of the eight-vertex model, the general equivalence of the two problems is by no means obvious. In fact, it was after considerable algebraic manipulation using a generalised weak-graph transformation that the equivalence was previously established [ l , 81. In this comment we present an alternative analysis of the eight-vertex model ro arrive at the same result. The new method is very simple and direct, and can be extended to other considerations. Consider a honeycomb lattice and draw bonds along its edges such that each edge is independently 'traced' or left 'open'. Then, there are eight different vertex configurations occurring at a vertex, which we show in figure 1. With each configuration we associate a vertex weight U, b, c or d and, as in [l] , we assume all weights to be positive. The partition function of the eight-vertex model is the generating function (1) 2 = Z(a, b, c, d ) =c unob"lc"2d"3 0 C C b b b C d Figure 1. Vertex configurations and weights for the symmetric eight-vertex model. 0305-4470/90/030375 +04%03.50 @ 1990 IOP Publishing Ltd 375