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

The limit n → ∞ of the classical O(n) ϕ4 model on a 3d film with free surfaces is studied. Its exact solution involves a self-consistent 1d Schrödinger equation, which is solved numerically for a partially discretized as well as for a fully discrete lattice model. Extremely precise results are obtained for the scaled Casimir force at all temperatures. Obtained via a single framework, they exhibit all relevant qualitative features of the thermodynamic Casimir force known from wetting experiments on 4He and Monte Carlo simulations, including a pronounced minimum below the bulk critical point