The critical Casimir effect in films for generic non-symmetry-breaking boundary conditions
Systems described by an O(n) symmetrical φ4 Hamiltonian areconsidered in ad-dimensional film geometry at their bulk critical points. Adetailed renormalization group (RG) study of the critical Casimir forces induced between the film’s boundary planes Bj,j = 1,2, by thermal fluctuations ispresented for the case where the O(n) symmetry remains unbroken by the surfaces. The boundary planes are assumed to cause short-range disturbancesof the interactions that can be modelled by standard surface contributions ∝φ2 corresponding to subcritical or critical enhancement of the surface interactions. This translates into mesoscopic boundary conditions of the genericsymmetry-preserving Robin type ∂nφ= ̊cjφ. RG-improved perturbation theory and Abel–Plana techniques are used to compute the L-dependent part fres of the reduced excess free energy per film area A → ∞ to two-loop order. When d < 4, it takes the scaling form fres ≈ D(c1LΦ/ν,c2LΦ/ν)/Ld−1 as L → ∞, where ci are scaling fields associated with the surface-enhancement variables c˚i, while Φ is a standard surface crossover exponent. The scaling function D(c1,c2) and its analogue for the Casimir force are determined via expansion in = 4 − d and extrapolated to d = 3 dimensions. In the special case c1 = c2 = 0, the expansion becomes fractional. Consistency with the known fractional expansions of D(0,0) and to order 3/2 is achieved by appropriate reorganization of RG-improved perturbation theory. For appropriate choices of c1 and c2, the Casimir forces can have either sign. Furthermore, crossovers from attraction to repulsion and vice versa may occur as L increases.
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