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The limit n to infinity of the classical O(n) phi^4 model on a 3d film withfree surfaces is studied. Its exact solution involves a self-consistent 1dSchr\"odinger equation, which is solved numerically for a partially discretizedas well as for a fully discrete lattice model. Numerically exact results areobtained for the scaled Casimir force at all temperatures. Obtained via asingle framework, they exhibit all relevant qualitative features of thethermodynamic Casimir force known from wetting experiments on Helium-4 andMonte Carlo simulations, including a pronounced minimum below the bulk criticalpoint.

Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces