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  "path": "/abs/2602.11292v1",
  "publishedAt": "2026-02-13T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Austen Fan",
    "Jin-Yi Cai",
    "Shuai Shao",
    "Zhuxiao Tang"
  ],
  "textContent": "**Authors:** Austen Fan, Jin-Yi Cai, Shuai Shao, Zhuxiao Tang\n\nWe 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.",
  "title": "New Planar Algorithms and a Full Complexity Classification of the Eight-Vertex Model"
}