The phase diagram of the 2D Ising model with anisotropic next-nearest-neighbor and four-spin interactions

In the present paper, the two-dimensional Ising model with anisotropic nearest-neighbor, next-nearest-neighbor and four-spin interactions has been studied. The ground states and energy of the model have been obtained. The model is equivalent to an eight-vertex model on its dual lattice. In some special cases, the model can be solved exactly as a zero-field eight-vertex model or a free-fermion model. Explicit phase diagrams are obtained exactly