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Recent exact $n\to\infty$ results for critical Casimir forces of the $O(n)$$\phi^4$ model on a three-dimensional strip bounded by two planar free surfacesat a distance $L$ are surveyed. This model has long-range order below the bulkcritical temperature $T_c$ if $L=\infty$, but remains disordered for all $T>0$when $L<\infty$. A proper analysis of its scaling behavior near $T_c$ is quitechallenging: Besides with bulk, boundary, and finite-size critical behaviors,one must deal with a nontrivial dimensional crossover. The model can be solvedexactly in the limit $n\to\infty$ in terms of the eigenvalues and eigenenergiesof a selfconsistent Schr\"odinger equation involving a potential $v(z)$ withthe near-boundary singular behavior $v(z\to 0+)\approx-1/(4z^2)+4m/(\pi^{2}z)$, where $m=1/\xi_+(|t|)$ is the inverse bulkcorrelation length and $t\sim (T-T_c)/T_c$, and a corresponding singularity atthe second boundary plane. The potential $v(z)$, the excess free energy, andthe Casimir force have been determined numerically with high precision. Exactanalytical results for a variety of properties such as series expansioncoefficients 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 behaviorof the residual free energy and the Casimir force can be obtained by acombination of boundary-operator and short-distance expansions, properextensions of inverse scattering theory, new trace formulae, and semiclassicalexpansions.

теоретическая и математическая физика

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 $n\to\infty$