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

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→∞ 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 bx^−(d+σ) as x→∞, with 2<σ<4 and 2<d+σ⩽6, on the Casimir effect at and near the bulk critical temperature T_c,∞, for 2<d<4. These interactions decay sufficiently fast to leave bulk critical exponents and other universal bulk quantities unchanged—i.e., they are irrelevant in the renormalization group (RG) sense. Yet they entail important modifications of the standard scaling behavior of the excess free energy and the Casimir force F_C. We generalize the phenomenological scaling Ansätze for these quantities by incorporating these long-range interactions. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form L^dF_C∕k_BT≈Ξ₀(L∕ξ_∞)+g_ωL^−ωΞ_ω(L∕ξ_∞)+g_σL^−ω_σΞ_σ(L∕ξ_∞). Here Ξ₀, Ξ_ω, and Ξ_σ are universal scaling functions; g_ω and g_σ are scaling fields associated with the leading corrections to scaling and those of the long-range interaction, respectively; ω and ω_σ=σ+η−2 are the associated correction-to-scaling exponents, where η denotes the standard bulk correlation exponent of the system without long-range interactions; ξ_∞ is the (second-moment) bulk correlation length (which itself involves corrections to scaling). The contribution ∝g_σ decays for T≠T_c,∞ 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 proportional to 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.