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

Systems described by $n$-component ${\varphi }^4$ models in a $\infty^{d-1}\times L$slabgeometryoffinitethickness$L$ are considered at and above their bulk critical temperature $T_{c,\mathrm{\infty }}$. 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 $T_{c,\mathrm{\infty }}$. The leading contributions to the critical Casimir amplitudes ${\mathrm{\Delta }}_{\mathrm{p}\mathrm{e}\mathrm{r}}$ and ${\mathrm{\Delta }}_{\mathrm{s}\mathrm{p},\mathrm{s}\mathrm{p}}$ beyond two-loop order are $\sim (u^{\ast }{)}^{(3-ϵ)\mathrm{/}2}$, where $u^{\ast }=O(ϵ)$ is the value of the renormalized ${\varphi }^4$ coupling at the infrared-stable fixed point. Besides integer powers of $ϵ$, the small-$ϵ$ expansions of these amplitudes involve fractional powers ${ϵ}^{k\mathrm{/}2}$, with $k\ge 3$, and powers of $\ln ϵ$. Explicit results to order ${ϵ}^{3\mathrm{/}2}$ are presented for ${\mathrm{\Delta }}_{\mathrm{p}\mathrm{e}\mathrm{r}}$ and ${\mathrm{\Delta }}_{\mathrm{s}\mathrm{p},\mathrm{s}\mathrm{p}}$, which are used to estimate their values at $d=3$.