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

Abstract

We consider systems confined to a $d$-dimensional slab of macroscopic lateral extension and finite thickness $L$ that undergo a continuous bulk phase transition in the limit $L\to\infty$ 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 $b x^{-(d+σ)}$ as $x\to\infty$, with $2<σ<4$ and $2<d+σ\leq 6$, 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 $L^{d} {\mathcal F}_C/k_BT \approx Ξ_0(L/ξ_\infty) + g_ωL^{-ω}Ξω(L/ξ_\infty) + g_σL^{-ω_\sigm a} Ξ_σ(L ξ_\infty)$. The contribution $\propto g_σ$ 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 $d+σ=6$, 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 $ω=ω_σ=4-d$ that occurs for the spherical model when $d+σ=6$, in conjunction with the $b$ dependence of $g_ω$.