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.

翻译：本文研究两态自旋系统配分函数的计算逼近性问题。该问题由一个$2\times 2$对称矩阵参数化。以往关于该问题的研究结果要么局限于矩阵元素非负的情形，要么局限于对角元相等的伊辛模型情形。本文研究推广至具有任意实值元素的$2\times 2$相互作用矩阵的情况。我们证明在参数空间的某些区域，甚至确定配分函数的符号都是#P-难问题；而在其他区域则存在配分函数的完全多项式逼近方案。我们的结果揭示了若干新的计算相变现象。