Excess free energy and Casimir forces in systems with long-range interactions of van-der-Wa.pdfVIP

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Excess free energy and Casimir forces in systems with long-range interactions of van-der-Wa.pdf

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Excess free energy and Casimir forces in systems with long-range interactions of van-der-Wa

a r X i v : c o n d - m a t / 0 5 1 0 4 0 5 v 2 [ c o n d - m a t .s t a t - m e c h ] 3 0 J a n 2 0 0 6 Excess free energy and Casimir forces in systems with long-range interactions of van-der-Waals type: General considerations and exact spherical-model results Daniel Dantchev,1, 2, ? H. W. Diehl,1 and Daniel Gru?neberg1, ? 1Fachbereich Physik, Universita?t Duisburg-Essen, Campus Essen, 45117 Essen, Germany 2Institute of Mechanics—Bulgarian Academy of Sciences, Academic Georgy Bonchev Street building 4, 1113 Sofia, Bulgaria (Dated: February 2, 2008) 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 Tc,∞, 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 FC . We generalize the phenomenological scaling ansa?tze for these quantities by incorporating these long-range interactions. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form Ld FC/kBT ≈ Ξ0 ( L/ξ∞ ) + gω L ?ωΞω ( L/ξ∞ ) + gσ L ?ωσ Ξσ ( L/ξ∞ ) . Here Ξ0, Ξω, and Ξσ are universal scaling functions; gω and gσ are scaling fields associated with the leading corrections to scaling and those of the long-range interaction, respectively; ω and ωσ = σ + η ? 2 are the associated correction-to-scaling exponents, where η denotes the standard bulk correlation exponent of the system without lon