An exactly solvable modification of the planar Ising ferromagnet is proposed which has a roughening transition below the Curie temperature. The computation confirms the de Gennes-Fisher scaling theory of correlations with homogeneous surface fields, giving an exponent value A t = 2.PACS numbers: 05.50.+q, 75.10.HkThe nature of the interface, or domain wall, between oppositely magnetized phases of a ferromagnet below its Curie temperature T c has been the subject of considerable recent interest. 1 " 4 It has been realized that, even though there exists a well-defined specific incremental free energy for a domain wall, 5 ' 6 in many situations the actual structure of such a wall is averaged out by capillary fluctuations unless some external stabilizing force is applied.A phenomenology which is generally accepted is that the domain wall may undergo large fluctuations on a length scale determined by the area of the interface, but carries with it a local structure which, in the critical region, varies on the scale of the correlation length. It is to this local structure that the successful phenomenological theory developed by van der Waals, by Cahn and Hilliard, and by Fisk and Widom 7 refers.The following remarks relate to simple-cubic Ising ferromagnets in d dimensions. For d =2, the interface between phases with magnetization , ±m*, ra* being the spontaneous magnetization, is always diffuse for 0T>T R the interface is diffuse. A roughening transition is said to occur at T R . The evidence for this, which is not beyond dispute, deprives from series expansions 9 and Monte Carlo simulations. 10 A rigorous proof that there exists a T R

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An exact statistical mechanical derivation is given of the critical Casimir forces for Ising strips with arbitrary surface fields applied to edges. Our results show that the strength as well as the sign of the force can be controled by varying the temperature or the fields. An interpretation of the results is given in terms of a linked cluster expansion. This suggests a systematic approach for deriving the critical Casimir force which can be used in more general models.