Complexity Classification of the Eight-Vertex Model

We prove a complexity dichotomy theorem for the eight-vertex model. For every setting of the parameters of the model, we prove that computing the partition function is either solvable in polynomial time or #P-hard. The dichotomy criterion is explicit. For tractability, we find some new classes of problems computable in polynomial time. For #P-hardness, we employ Möbius transformations to prove the success of interpolations. Computer Sciences Department, University of Wisconsin, Madison, USA. [email protected] School of Mathematics, Jilin University, Changchun, China. [email protected]