On the equivalence of the Ising model with a vertex problem?

The planar Ising model in an external magnetic field and with multi-spin interactions is shown to be equivalent to a vertex problem. The equivalence can be extended to higher spins and higher dimensions. In a recent paper Malakis (1979) reported the proof of an equivalence between a certain staggered vertex model and a zero-field Ising model. The equivalence has since been extended (Malakis 1980) to a wider class of vertex problems on the square lattice. The purpose of this Letter is to point out that the result of Malakis (1979) is not new and, in fact, can be easily derived in a very general form to include the Ising model with non-zero magnetic field as well as other lattices. It can also be extended to higher spins and higher dimensions. In view of the rather elaborate effort required in Malakis' proof, it appears desirable to report on this development. It was first noted by Wu and Lin (1975) that the square lattice Ising model in a non-zero magnetic field is reducible to a staggered vertex model. This equivalence, of which the result of Malakis (1979) is a special case, has also been reported in Wu (1978). The derivation given by Wu and Lin (1975) is very simple. We now explore its extensions. Consider an even planar graph (or lattice) G which has an even number of edges incident at each vertex. Examples are the square, KagomC and triangular lattices. Since the faces of G are bipartite, we may shade one set of the faces as shown in figure 1. Consider next an Ising model whose spins are located at the shaded faces of G and whose interactions, including external magnetic field and possibly multi-spin interactions, are decomposable into sums over the vertices of G. Specifically, let v1 . . . , (+k be the spins surrounding a site of G. Then the Ising Hamiltonian 2 takes the form Under this circumstance, we have the following equivalence: the Ising model (1) is equivalent to a vertex model on G. The proof of this equivalence, which also serves to define the vertex model, is extremely simple. For a given spin configuration in the Ising model, encircle the (shaded) faces 'r Supported in part by NSF Grant No DMR 78-18808. 0305-4470/80/090303 + 03$01.50 @ 1980 The Institute of Physics L303 L304 Letter to the Editor