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 in the case of a zero magnetic field and in an imaginary magnetic field i(n/2)kBT. 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 the zero field and one soluble case in the imaginary field are examined. In the case that the free-fermion condition holds in zero field, we find that the ground states in the low temperature limit include the configurations disobeying the ice rules. In another case in zero field where the four-spin interactions are -8 and another case in imaginary field where 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 to the latter two cases, we have shown alternative approaches to the residual entropy problem of square ice.