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

and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as

b x^{-(d+\sigma)}

as

x\to\infty

, with

2<\sigma<4

and

2<d+\sigma\leq 6

, on the 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-sectional area, 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 hence becomes dominant in an appropriate regime of temperatures and

L

. We derive exact results for spherical and Gaussian models which confirm these findings. In the case

d+\sigma =6

, which includes that of nonretarded van-der-Waals interactions 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 originate from the degeneracy