Residual Entropy of a Two-dimensional Ising Model with Crossing and Four-spin Interactions

We study the residual entropy of a two-dimensional Ising model with crossing and four-spin interactions, both for the case that in zero magnetic field and that in an imaginary magnetic field i({\pi}/2)kT. The spin configurations of this Ising model can be mapped into the hydrogen configurations of square ice with the defined standard direction of the hydrogen bonds. Making use of the equivalence of this Ising system with the exactly solved eight-vertex model and taking the low temperature limit, we obtain the residual entropy. Two soluble cases in zero field and one soluble case in imaginary field are examined. In the case that the free-fermion condition holds in zero field, we find the ground states in low temperature limit include the configurations disobeying the ice rules. In another case in zero field that the four-spin interactions are -{\infty}, and the case in imaginary field that the four-spin interactions are 0, the residual entropy exactly agrees with the result of square ice determined by Lieb in 1967. In the solutions of the latter two cases, we have shown alternative approaches to the residual entropy problem of square ice.