The three-dimensional O(n) ϕ 4 model on a strip with free boundary conditions: Exact results for a nontrivial dimensional crossover in the limit n→∞

Abstract

We briefly review recent results of exact calculations of critical Casimir forces of the O(n) ϕ4 model as n→∞ 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 Tc of the bulk phase transition only in the limit L→∞ but remains disordered for all T > 0 for an arbitrary finite strip width L /< 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.