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The $O(n)$ ${\phi}^4$ model on a slab $\mathbb{R}^{d-1}\times[0,L]$ boundedby free surfaces is studied for $2<d<4$ in the limit $n\to\infty$. Theself-consistent potential $V(z)$ which the exact $n\to\infty$ solution of themodel involves is analysed by means of boundary operator expansions. Buildingon the known exact $n\to\infty$ solution for $V(z)$ in the semi-infinite case$L=\infty$ at the bulk critical point, we exactly determine two types ofcorrections to this potential: (i) those linear in the temperature scalingfield $t$ at $L=\infty$, and (ii) the leading $L$-dependent (distant-wall)corrections at the critical point. From (i) exact analytical results at $d=3$are obtained for the leading temperature singularity of the excess surface freeenergy and the implied asymptotic behaviours of the scaling functions$\Theta_3(x)$ and $\vartheta_3(x)$ of the residual free energy $f_{\rm res}=L^{1-d}\,\Theta_d(tL)$ and the critical Casimir force $\beta\mathcal{F}_{\rmC}(T,L)=L^{-d}\,\vartheta_d(tL)$ in the limit $x\to 0\pm$. The secondderivative $\vartheta_3''(0)$ is computed exactly.

journal of physics. a, mathematical and theoretical

The O(n) $\phi^4$ model with free surfaces in the large-$n$ limit: Some  exact results for boundary critical behaviour, fluctuation-induced forces and  distant-wall corrections