Exact results of an Ising model with three- and five-spin interactions on a square lattice

We investigate an Ising model with three-spin interactions and five-spin interactions on a square lattice. We use exact mapping relations among interaction parameters and the temperature. We found that the system is equivalent to the eight-vertex model in some cases. For the system with an imaginary interaction parameter, the system is reduced to the exactly solvable Ising model with nearest-neighbor two-spin interactions and some non-nearest-neighbor two-spin interactions, the so-called Vaks–Larkin–Ovchinnikov model.