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 $L\ensuremath{\rightarrow}\ensuremath{\infty}$. The effective forces induced by thermal fluctuations at and above the bulk critical temperature ${T}_{c,\ensuremath{\infty}}$ (thermodynamic Casimir effect) are investigated below the upper critical dimension ${d}^{*}=4$ 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 [Europhys. Lett. 75, 241 (2006)], the zero modes that are present in Landau theory at ${T}_{c,\ensuremath{\infty}}$ make conventional renormalization-group-improved perturbation theory in $4\ensuremath{-}ϵ$ 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 $T\ensuremath{\geqslant}{T}_{c,\ensuremath{\infty}}$ as functions of $\mathsf{L}\ensuremath{\equiv}L∕{\ensuremath{\xi}}_{\ensuremath{\infty}}$, where ${\ensuremath{\xi}}_{\ensuremath{\infty}}$ is the bulk correlation length. Scaling functions of the $L$-dependent residual free energy per area are obtained, whose $\mathsf{L}\ensuremath{\rightarrow}0$ limits are in conformity with previous results for the Casimir amplitudes ${\ensuremath{\Delta}}_{C}$ to $O({ϵ}^{3∕2})$ and display a more reasonable small-$\mathsf{L}$ behavior inasmuch as they approach the critical value ${\ensuremath{\Delta}}_{C}$ monotonically as $\mathsf{L}\ensuremath{\rightarrow}0$. Extrapolations to $d=3$ for the Ising case $n=1$ with periodic boundary conditions are in fair agreement with Monte Carlo results. In the case of special-special boundary conditions, extrapolations to $d=3$ are hampered by the fact that the one-loop result for the inverse finite-size susceptibility becomes negative for some values of $\mathsf{L}$ when $ϵ\ensuremath{\gtrsim}0.83$.