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

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 Möbius transformation from complex analysis. The proof also makes use of cyclotomic fields.

💡 Research Summary

This paper establishes a complete complexity classification for the eight‑vertex model on planar graphs. The eight‑vertex model is defined on 4‑regular graphs; each vertex may be in one of eight local orientation types, each weighted by a (possibly complex) Boltzmann factor w₁,…,w₈. The partition function Z₈V(G)=∑{σ∈O(G)}∏{i=1}^8 w_i^{n_i} sums over all even orientations (every vertex has even indegree). While a dichotomy for general graphs was known (Cai et al., 2015), the planar case remained open.

The authors embed the model into the Holant framework as Holant(=₂ | f) where the 4‑ary signature f has matrix

M(f)=⎡a 0 0 b; 0 c d 0; 0 w z 0; y 0 0 x⎤.

The outer sub‑matrix (a b y x) and the inner sub‑matrix (c d w z) are the main objects of analysis. They classify parameter settings according to the pattern of zero entries, defining “zero pairs” (two entries that are simultaneously zero). The number N of zero entries leads to two major cases:

• Case I (N ≥ 2) – at least two zero pairs. This case splits into symmetric and asymmetric conditions. In the symmetric condition the parameters satisfy a=b, c=d, etc., yielding a highly symmetric weight vector (a,a,b,b,c,c,d,d). The authors show that under this symmetry the eight‑vertex model on a planar graph is equivalent to the planar Even‑Coloring problem: count edge‑colorings with two colors such that each vertex has an even number of incident green edges. This equivalence is realized by a combinatorial gadget that preserves the Holant value.

  After the reduction, they apply a holographic transformation H = (1/√2)