The three-dimensional $O(n)$ $\phi^4$ model on a strip with free boundary conditions: Exact results for a nontrivial dimensional crossover in the limit $n\to\infty$

We briefly review recent results of exact calculations of critical Casimir forces of the $O(n)$ $\phi^4$ model as $n\to\infty$ on a three-dimensional strip bounded by two planar free surfaces at a distance $L$. This model has long-range order below the critical temperature $T_{\mathrm c}$ of the bulk phase transition only in the limit $L\to\infty$ but remains disordered for all $T>0$ for an arbitrary finite strip width $L<\infty$. A proper description of the system scaling behavior near $T_{\mathrm c}$ turns out to be a quite challenging problem because in addition to bulk, boundary, and finite-size critical behaviors, a nontrivial dimensional crossover must be handled. The model admits an exact solution in the limit $n\to\infty$ in terms of the eigenvalues and eigenenergies of a self-consistent Schrödinger equation. This solution contains a potential $v(z)$ with the near-boundary singular behavior $v(z\to0+)\approx-1/(4z^2)+4m/(\pi^2z)$, where $m=1/\xi_+(|t|)$ is the inverse bulk correlation length and $t\sim(T-T_{\mathrm c})/T_{\mathrm c}$, and a corresponding singularity at the second boundary plane. In recent joint work with colleagues, the potential $v(z)$, the excess free energy, and the Casimir force were obtained numerically with high precision. We explain how these numerical results can be complemented by exact analytic ones for several quantities (series expansion coefficients of $v(z)$, the scattering data of $v(z)$ in the semi-infinite case $L=\infty$ for all $m\gtreqless 0$, and the low-temperature asymptotic behavior of the residual free energy and the Casimir force) by a combination of boundary-operator and short-distance expansions, proper extensions of the inverse scattering theory, new trace formulas, and semiclassical expansions.