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Kasteleyn's theorem may be generalized to allow complex phase factors in the weighted adjacency matrix: for a transition cycle passing through sites 1,2,...,2n, the phase factors must satisfy η12 η34... η2n-1,2n =- η23 η45... η2n,1. Complex phase factors provide a more elegant solution of the square lattice dimer model (Ref.). However, they do not help in the case of the kagome lattice (Ref.) or TKL; we have found that any orientation with the periodicity of the original lattice violates the generalized Kasteleyn theorem even if the phase factors are allowed to be arbitrary complex numbers.
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