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We stress that this is a term-by-term correspondence in the sense that, given the appropriate basis, the squared amplitudes of the basis states comprising the ground state are identical to the Gibbs weights of the corresponding statistical mechanical states (up to an overall constant).
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We stress that this is a term-by-term correspondence in the sense that, given the appropriate basis, the squared amplitudes of the basis states comprising the ground state are identical to the Gibbs weights of the corresponding statistical mechanical states (up to an overall constant).
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In a fully packed case, there is an additional geometric constraint limiting these winding numbers to either only even or only odd, depending on the size of the torus.
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This derivation can be made slightly more formal if one defines a joint measure on both spin and bond configurations: μ (σ,ω) = vb (ω) Δ (σ,ω). This is known as Edwards-Sokal measure (Ref.). As follows from Eq. 12, when traced over both spins and bond occupations, it gives the desired partition function. Its marginal with respect to bond configurations is the Gibbs weight for spins, while its marginal with respect to spin configurations defines the weight for random clusters.
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It is also known as a cyclomatic number and for the case of a planar graph is equal to the number of finite faces (Ref.).
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