Abstract
A new extensive variable A and its conjugate field II are introduced for the Kasteleyn dimer model on the honeycomb lattice via an isomorphic membrane-polymer chain model. The resultant phase diagram has a first-order phase line, thereby revealing a non-local order parameter in this model for the first time. The first-order line terminates in a critical point with critical exponents α=α=0, β=1=γ=γ, and δ=2. The 32-order exponents appear on the line Π=Πcritical.

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