Critical surface of the Blume-Emery-Griffiths model on the honeycomb lattice

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Abstract
We consider the Blume-Emergy-Griffiths (BEG) model on the honeycomb lattice and obtain a closed-form expression for the critical surface of second-order transitions. The BEG model is first formulated as a three-state vertex model. Using the fact that the BEG critical surface coincides with that of a general three-state vertex model, we construct critical surfaces by forming polynomial combinations of vertex weights that are invariant under an O(3) gauge transformation. We then carry out a finite-size analysis of the BEG model, and use data so obtained to determine coefficients appearing in the polynomial combination. This procedure leads to a closed-form expression of the critical surface which reproduces all numerical data accurately.

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