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We study two $Q$-state Potts models coupled by the product of their energyoperators, in the regime $2 < Q \le 4$ where the coupling is relevant. Aparticular choice of weights on the square lattice is shown to be equivalent tothe integrable $a_3^{(2)}$ vertex model. It corresponds to a selfdual system oftwo antiferromagnetic Potts models, coupled ferromagnetically. We derive theBethe Ansatz equations and study them numerically for two arbitrary twistangles. The continuum limit is shown to involve two compact bosons and one noncompact boson, with discrete states emerging from the continuum at appropriatetwists. The non compact boson entails strong logarithmic corrections to thefinite-size behaviour of the scaling levels, the understanding of which allowsus to correct an earlier proposal for some of the critical exponents. Inparticular, we infer the full set of magnetic scaling dimensions (watermelonoperators) of the Potts model.

Non compact continuum limit of two coupled Potts models