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We consider systems confined to a $d$-dimensional slab of macroscopic lateralextension and finite thickness $L$ that undergo a continuous bulk phasetransition in the limit $L\to\infty$ and are describable by an O(n) symmetricalHamiltonian. Periodic boundary conditions are applied across the slab. We studythe effects of long-range pair interactions whose potential decays as $bx^{-(d+\sigma)}$ as $x\to\infty$, with $2<\sigma<4$ and $2<d+\sigma\leq 6$, onthe Casimir effect at and near the bulk critical temperature $T_{c,\infty}$,for $2<d<4$. For the scaled reduced Casimir force per unit cross-sectionalarea, we obtain the form $L^{d} {\mathcal F}_C/k_BT \approx \Xi_0(L/\xi_\infty)+ g_\omega L^{-\omega}\Xi\omega(L/\xi_\infty) + g_\sigma L^{-\omega_\sigm a}\Xi_\sigma(L \xi_\infty)$. The contribution $\propto g_\sigma$ decays for$T\neq T_{c,\infty}$ algebraically in $L$ rather than exponentially, and hencebecomes dominant in an appropriate regime of temperatures and $L$. We deriveexact results for spherical and Gaussian models which confirm these findings.In the case $d+\sigma =6$, which includes that of nonretarded van-der-Waalsinteractions in $d=3$ dimensions, the power laws of the corrections to scaling$\propto b$ of the spherical model are found to get modified by logarithms.Using general RG ideas, we show that these logarithmic singularities originatefrom the degeneracy $\omega=\omega_\sigma=4-d$ that occurs for the sphericalmodel when $d+\sigma=6$, in conjunction with the $b$ dependence of $g_\omega$.Comment: 28 RevTeX pages, 12 eps figures, submitted to PR

physical review. e, statistical, nonlinear and soft matter physics

Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: General considerations and exact spherical-model results