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 $n\to\infty$

Abstract

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 surfaces at a distance $L$ are surveyed. This model has long-range order below the bulk critical 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 quite challenging: Besides with bulk, boundary, and finite-size critical behaviors, one must deal with a nontrivial dimensional crossover. The model can be solved exactly in the limit $n\to\infty$ in terms of the eigenvalues and eigenenergies of a selfconsistent Schr\"odinger equation involving a potential $v(z)$ with the near-boundary singular behavior $v(z\to 0+)\approx -1/(4z^2)+4m/(\pi^{2}z)$, where $m=1/\xi_+(|t|)$ is the inverse bulk correlation length and $t\sim (T-T_c)/T_c$, and a corresponding singularity at the second boundary plane. The potential $v(z)$, the excess free energy, and the Casimir force have been determined numerically with high precision. Exact analytical results for a variety of properties such as 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 can be obtained by a combination of boundary-operator and short-distance expansions, proper extensions of inverse scattering theory, new trace formulae, and semiclassical expansions.