Casimir force in O(n) systems with a diffuse interface

We study the behavior of the Casimir force in O(n) systems with a diffuse interface and slab geometry ∞^d−1×L, where 2<d<4 is the dimensionality of the system. We consider a system with nearest-neighbor anisotropic interaction constants J_∥ parallel to the film and J_⊥ across it. We argue that in such an anisotropic system the Casimir force, the free energy, and the helicity modulus will differ from those of the corresponding isotropic system, even at the bulk critical temperature, despite that these systems both belong to the same universality class. We suggest a relation between the scaling functions pertinent to the both systems. Explicit exact analytical results for the scaling functions, as a function of the temperature T, of the free energy density, Casimir force, and the helicity modulus are derived for the n→∞ limit of O(n) models with antiperiodic boundary conditions applied along the finite dimension L of the film. We observe that the Casimir amplitude Δ_Casimir(d∣J_⊥,J_∥) of the anisotropic d-dimensional system is related to that of the isotropic system Δ_Casimir(d) via Δ_Casimir(d∣J_⊥,J_∥)=(J_⊥∕J_∥)^(d−1)∕2Δ_Casimir(d). For d=3 we derive the exact Casimir amplitude Δ_Casimir(3,∣J_⊥,J_∥)=[Cl₂(π∕3)∕3−ζ(3)∕(6π)](J_⊥∕J_∥), as well as the exact scaling functions of the Casimir force and of the helicity modulus Υ(T,L). We obtain that β_cΥ(T_c,L)=(2∕π²)[Cl₂(π∕3)∕3+7ζ(3)∕(30π)](J_⊥∕J_∥)L⁻¹, where T_c is the critical temperature of the bulk system. We find that the contributions in the excess free energy due to the existence of a diffuse interface result in a repulsive Casimir force in the whole temperature region.