Complexity Classification of the Eight-Vertex Model

TL;DR

This paper establishes a clear complexity classification for the eight-vertex model's partition function, identifying conditions under which it is either efficiently computable or P-hard, with explicit criteria and new polynomial-time solvable classes.

Contribution

It provides the first explicit complexity dichotomy theorem for the eight-vertex model, including new polynomial-time solvable classes and a novel proof technique for hardness.

Findings

Partition function is either polynomial-time computable or P-hard for all parameter settings.

Explicit criteria for the complexity classification are provided.

New classes of problems are identified as polynomial-time solvable.

Abstract

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 M\"{o}bius transformations to prove the success of interpolations.