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 ${\varphi }^4$ 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 $\sim (u^{\ast }{)}^{(3-\epsilon )/2}$, where $u^{\ast }=O(\epsilon )$ is the value of the renormalized ${\varphi }^4$ 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 $k\ge 3$, 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 $d=3$.

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