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

Abstract: The $O(n)$ ${\phi}^4$ model on a slab $\mathbb{R}^{d-1}\times[0,L]$ bounded by free surfaces is studied for $2<d<4$ in the limit $n\to\infty$. The self-consistent potential $V(z)$ which the exact $n\to\infty$ solution of the model involves is analysed by means of boundary operator expansions. Building on 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 of corrections to this potential: (i) those linear in the temperature scaling field $t$ 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 free energy 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 $\beta\mathcal{F}_{\rm C}(T,L)=L^{-d}\,\vartheta_d(tL)$ in the limit $x\to 0\pm$. The second derivative $\vartheta_3''(0)$ is computed exactly.