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

Abstract

Systems described by $n$-component $\phi^4$ models in a $\infty^{d-1}\times L$ slab geometry of finite thickness $L$ are considered at and above their bulk critical temperature $T_{c,\infty}$. The renormalization-group improved perturbation theory commonly employed to investigate the fluctuation-induced forces (``thermodynamic Casimir effect'') in $d=4-\epsilon$ 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-$\epsilon$ expansion results which remains well behaved down to $T_{c,\infty}$. The leading contributions to the critical Casimir amplitudes $\Delta_{\mathrm{per}}$ and $\Delta_{\mathrm{sp},\mathrm{sp}}$ beyond two-loop order are $\sim (u^*)^{(3-\epsilon)/2}$, where $u^*=O(\epsilon)$ is the value of the renormalized $\phi^4$ coupling at the infrared-stable fixed point. Besides integer powers of $\epsilon$, the small-$\epsilon$ expansions of these amplitudes involve fractional powers $\epsilon^{k/2}$, with $k\geq 3$, and powers of $\ln \epsilon$. Explicit results to order $\epsilon^{3/2}$ are presented for $\Delta_{\mathrm{per}}$ and $\Delta_{\mathrm{sp},\mathrm{sp}}$, which are used to estimate their values at $d=3$.