Thermodynamic Casimir effect in the large-$n$ limit: Exact results for slabs with free surfaces

The $O(n)$ $\phi^4$-model for a three-dimensional slab of thickness $L$ and infinite lateral extension is investigated in the large-$n$ limit. The effective (Casimir-like) forces that are induced between the two confining boundary planes by thermodynamic fluctuations in such systems at and near bulk criticality are studied. While systems with periodic or antiperiodic boundary conditions perpendicular to the planes are translationally invariant and thus can be treated analytically, the physical relevant case of free boundary conditions leads to a breaking of translational invariance perpendicular to the boundary planes. In the large-$n$ limit one arrives at a spherical model with separate constraints for each layer parallel to the confining surfaces. The resulting self-consistent Schr\"odinger-type equation is solved numerically at and near the bulk critical temperature to obtain the large-$n$ limit of the scaling functions of the Casimir force and the excess free energy. The Casimir amplitude is calculated with high accuracy to take the value $\Delta_{\mathrm{C}}=-0.01077340685025(10)$. The Casimir force scaling function shows a minimum below the bulk critical temperature similar to the $n=2$ result.