  We study the approximability of computing the partition functions of
two-state spin systems. The problem is parameterized by a $2\times 2$ symmetric
matrix. Previous results on this problem were restricted either to the case
where the matrix has non-negative entries, or to the case where the diagonal
entries are equal, i.e. Ising models. In this paper, we study the
generalization to arbitrary $2\times 2$ interaction matrices with real entries.
We show that in some regions of the parameter space, it's \#P-hard to even
determine the sign of the partition function, while in other regions there are
fully polynomial approximation schemes for the partition function. Our results
reveal several new computational phase transitions.
