Thermodynamic Casimir effects involving interacting field theories with zero modes

Systems with an O(n) symmetrical Hamiltonian are considered in a d-dimensional slab geometry of macroscopic lateral extension and finite thickness L that undergo a continuous bulk phase transition in the limit L→∞. The effective forces induced by thermal fluctuations at and above the bulk critical temperature T_c,∞ (thermodynamic Casimir effect) are investigated below the upper critical dimension d^*=4 by means of field-theoretic renormalization-group methods for the case of periodic and special-special boundary conditions, where the latter correspond to the critical enhancement of the surface interactions on both boundary planes. As shown previously [ Europhys. Lett. 75 241 (2006)], the zero modes that are present in Landau theory at T_c,∞ make conventional renormalization-group-improved perturbation theory in 4−ϵ dimensions ill-defined. The revised expansion introduced there is utilized to compute the scaling functions of the excess free energy and the Casimir force for temperatures T⩾T_c,∞ as functions of L≡L∕ξ_∞, where ξ_∞ is the bulk correlation length. Scaling functions of the L-dependent residual free energy per area are obtained, whose L→0 limits are in conformity with previous results for the Casimir amplitudes Δ_C to O(ϵ^3∕2) and display a more reasonable small-L behavior inasmuch as they approach the critical value Δ_C monotonically as L→0. Extrapolations to d=3 for the Ising case n=1 with periodic boundary conditions are in fair agreement with Monte Carlo results. In the case of special-special boundary conditions, extrapolations to d=3 are hampered by the fact that the one-loop result for the inverse finite-size susceptibility becomes negative for some values of L when ϵ≳0.83.