Abstract
Suppose \(\varphi\) and \(\psi\) are two angles satisfying \(\tan(\varphi) = 2 \tan(\psi) > 0\). We prove that under this condition \(\varphi\) and \(\psi\) cannot be both rational multiples of π. We use this number theoretic result to prove a classification of the computational complexity of spin systems on k-regular graphs with general (not necessarily symmetric) real valued edge weights. We establish explicit criteria, according to which the partition functions of all such systems are classified into three classes: (1) Polynomial time computable, (2) #P-hard in general but polynomial time computable on planar graphs, and (3) #P-hard on planar graphs. In particular, problems in (2) are precisely those that can be transformed by a holographic reduction to a form solvable by the Fisher-Kasteleyn-Temperley algorithm for counting perfect matchings in a planar graph.
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Acknowledgement
We thank Kurt Kilpela for his contributions, especially for a conversation which triggered the discovery that incommensurability between tangent values and angles over \(\pi\) could be exploited. We thank Tonghai Yang for discussions on number theory. We also thank Yijia Chen, Pinyan Lu, Xiaoming Sun and Tyson Williams for insightful discussions. We thank the anonymous referees for their insightful comments, and we are particularly grateful for the anonymous referee who suggested an alternative proof of Theorem 2.2, which led us to search for the simplified proofs presented here.
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Cai, JY., Fu , Z., Girstmair, K. et al. A Complexity Trichotomy for k-Regular Asymmetric Spin Systems Using Number Theory. comput. complex. 32, 4 (2023). https://doi.org/10.1007/s00037-023-00237-w
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DOI: https://doi.org/10.1007/s00037-023-00237-w