Exact results for thermodynamics of the classical field theories: Sine-Gordon 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 493, 571 (1997); V. Fateev, S. Lukyanov, A. B. Zamolodchikov, and Al. B. Zamolodchikov, Phys. Lett. B 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-Garden model. [S0163-1829(99)00641-4].