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Fluctuation-induced forces in periodic slabs: Breakdown of ϵ expansion at the bulk critical point and revised field theory

Systems described by n-component ϕ4 models in a ∞d − 1×L slab geometry of finite thickness L are considered at and above their bulk critical temperature Tc,∞. The renormalization-group improved perturbation theory commonly employed to investigate the fluctuation-induced forces (“thermodynamic Casimir effect”) in d = 4 − ϵ bulk dimensions is re-examined. It is found to be ill-defined beyond two-loop order because of infrared singularities when the boundary conditions are such that the free propagator in slab geometry involves a zero-energy mode at bulk criticality. This applies to periodic boundary conditions and the special-special ones corresponding to the critical enhancement of the surface interactions on both confining plates. The field theory is reorganized such that a small-ϵ expansion results which remains well behaved down to Tc,∞. The leading contributions to the critical Casimir amplitudes Δper and Δsp,sp beyond two-loop order are ∼ (u*)(3 − ϵ)/2, where u* = O(ϵ) is the value of the renormalized ϕ4 coupling at the infrared-stable fixed point. Besides integer powers of ϵ, the small-ϵ expansions of these amplitudes involve fractional powers ϵk/2, with k ⩾ 3, and powers of ln ϵ. Explicit results to order ϵ3/2 are presented for Δper and Δsp,sp, which are used to estimate their values at d = 3.