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Exact Results for Thermodynamics of the Classical Field Theories: Sine- and Sinh-Gordon Models

Using the recently obtained exact results for the expectation values of operators in the sine- and sinh-Gordon models [A. B. Zamolodchikov and S. Lukyanov, Nucl. Phys. B{\bf 493}, 571 (1997), V. Fateev, S. Lukyanov, A. B. Zamolodchikov and Al. B. Zamolodchikov, Phys. Lett. B{\bf 406}, 83 (1997)] we calculate the specific heat of the corresponding two dimensional Euclidean (classical) models. We show that the temperature dependence of the specific heat of the sine-Gordon model, in the commensurate phase, has a maximum well below the Kosterlitz-Thouless transition and that the sinh-Gordon model is thermodynamically unstable in the strong coupling regime. We give also the temperature dependence of the specific heat in the incommensurate phase of the sine-Gordon model.