摘要：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 \pi. 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 to a form solvable by the Fisher-Kasteleyn-Temperley algorithm by a holographic reduction.
关键词：Spin Systems; Holant Problems; Number Theory; Characters; Cyclotomic Fields
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