Thermodynamic Casimir effects involving interacting field theories with zero modes

Abstract

Systems with an O(n) symmetrical Hamiltonian are considered in a $d$-dimensional slab geometry of macroscopic lateral extension and finite thickness $L$ that undergo a continuous bulk phase transition in the limit $L\to \infty$. The effective forces induced by thermal fluctuations at and above the bulk critical temperature $T_{c,\infty }$ (thermodynamic Casimir effect) are investigated below the upper critical dimension $d^{\ast }=4$ by means of field-theoretic renormalization group methods for the case of periodic and special-special boundary conditions, where the latter correspond to the critical enhancement of the surface interactions on both boundary planes. As shown previously [\textit{Europhys. Lett.} \textbf{75}, 241 (2006)], the zero modes that are present in Landau theory at $T_{c,\infty }$ make conventional RG-improved perturbation theory in $4-\epsilon$ dimensions ill-defined. The revised expansion introduced there is utilized to compute the scaling functions of the excess free energy and the Casimir force for temperatures $T\geqT_{c,\infty}$ as functions of $\mathsf{L}\equiv L/{\xi }_{\infty }$, where ${\xi }_{\infty }$ is the bulk correlation length. Scaling functions of the $L$-dependent residual free energy per area are obtained whose $\mathsf{L}\to 0$ limits are in conformity with previous results for the Casimir amplitudes ${\Delta }_C$ to $O({\epsilon }^{3/2})$ and display a more reasonable small-$\mathsf{L}$ behavior inasmuch as they approach the critical value ${\Delta }_C$ monotonically as $\mathsf{L}\to 0$.

23 pages, 10 figures

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