The three-dimensional O(n) ϕ⁴ model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover in the limit n→∞

Original · EN

Recent exact n→∞ results for critical Casimir forces of the O(n) ϕ⁴ 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 Tc if L=∞, but remains disordered for all T>0 when L<∞. A proper analysis of its scaling behavior near Tc 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→∞ in terms of the eigenvalues and eigenenergies of a selfconsistent Schrödinger equation involving a potential v(z) with the near-boundary singular behavior v(z→ 0+)≈ -1/(4z²)+4m/(π²z), where m=1/ξ+(|t|) is the inverse bulk correlation length and t (T-Tc)/Tc, 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=∞ for all m 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.