New Planar Algorithms and a Full Complexity Classification of the Eight-Vertex Model

TL;DR

This paper provides a complete complexity classification of the planar eight-vertex model, identifying when its partition function is efficiently computable or P-hard, and introduces new polynomial-time solvable cases beyond known algorithms.

Contribution

It establishes a full complexity classification for the planar eight-vertex model, including new polynomial-time solvable models using combinatorial and holographic transformations.

Findings

Complete classification of the eight-vertex model complexity

Discovery of new P-time solvable models beyond Kasteleyn's algorithm

Connections to Ising model, conformal interpolation, and complex analysis

Abstract

We prove a complete complexity classification theorem for the planar eight-vertex model. For every parameter setting in ${\mathbb C}$ for the eight-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) \#P-hard for general graphs but computable in P-time for planar graphs, or (3) \#P-hard even for planar graphs. The classification has an explicit criterion. In (2), we discover new P-time computable eight-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. They are obtained by a combinatorial transformation to the planar {\sc Even Coloring} problem followed by a holographic transformation to the tractable cases in the planar six-vertex model. In the process, we also encounter non-local connections between the planar eight vertex model and the bipartite Ising model, conformal lattice interpolation and M\"{o}bius transformation from complex analysis. The proof also makes use of cyclotomic fields.