Exact Solution of a Triangular Ising Model in a Nonzero Magnetic Field 1

One outstanding unsolved problem in statistical mechanics is the closedform computation of the free energy of the two-dimensional Ising model in a nonzero magnetic field. In 1976 Verhagen (1~ considered one particular triangular Ising model, and obtained its solution along a certain trajectory in the parameter space. This solution, which was obtained through the consideration of a stochastic crystal growth model, has since been extended by Rujim (~ to the fully isotropic antiferromagnetic model with nearestneighbor interactions. Quite recently, it has been further recognized that the nonzero field triangular Ising model is related to a number of other important two-dimensional lattice-statistical problems. The nearestneighbor model is shown to relate to the problem of directed lattice animals, (~ and the Ising model with twoand three-spin interactions is equivalent to cellular automata and directed percolation. (4) Therefore, it is not without interest to seek for further solutions of the triangular Ising system. In this connection it should be pointed out that the hard-hexagon problem solved by Baxter (5~ corresponds to an infinite-field, infinite-