Europhys. Lett.
Volume 75, Number 2, July 2006
Page(s) 241 - 247
Section General
Published online 14 June 2006
Europhys. Lett., 75 (2), pp. 241-247 (2006)
DOI: 10.1209/epl/i2006-10090-0

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

H. W. Diehl1, Daniel Grüneberg1 and M. A. Shpot1, 2

1  Fachbereich Physik, Universität Duisburg-Essen - Campus Essen D-45117 Essen, Germany
2  Institute for Condensed Matter Physics - 79011 Lviv, Ukraine

received 2 May 2006; accepted in final form 23 May 2006
published online 14 June 2006

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.

05.70.Jk - Critical point phenomena.
68.15.+e - Liquid thin films.
11.10.-z - Field theory.

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