Exact solution for the order parameter profiles and the Casimir force in 4He superfluid films in an effective field theory

We present an analytical solution of an effective field theory which, in one of its formulations, is equivalent to the Ginzburg’s Ψ -theory for the behavior of the Casimir force in a film of 4 He in equilibrium with its vapor near the superfluid transition point. We consider three versions of the theory, depending on the way one determines its parameters from the experimental measurements. We present exact results for the behavior of the order parameter profiles and of the Casimir force within this theory, which is characterized by d = 3 , ν = 2 ∕ 3 and β = 1 ∕ 3 , where d is the bulk spatial dimension and ν and β are the usual critical exponents. In addition, we revisit relevant experiments (Garcia and Chan, 1999) and (Ganshin et al., 2006) in terms of our findings. We find reasonably good agreement between our theoretical predictions and the experimental data. We demonstrate analytically that our calculated force is attractive. The position of the extremum is predicted to be at x min = π , with x = ( L ∕ ξ 0 ) ( T ∕ T λ − 1 ) 1 ∕ ν , which value effectively coincides with the experimental finding x min = 3 . 2 ± 0 . 18 . Here L is the thickness of the film, T λ is the bulk critical temperature and ξ 0 is the correlation length amplitude of the system for temperature T > T λ . The theoretically predicted position of the minimum does not depend on the one adjustable parameter, M , entering the theory. The situation is different with respect to the largest absolute value of the scaling function, which depends on both ξ 0 and on M . For this value the experiments yield − 1 . 30 . If one uses ξ 0 = 1 . 63 A , as in the original Ψ theory of Ginsburg, one obtains the closest approach to the experimental value − 1 . 848 with M = 0 . If one uses M = 0 . 5 , as inferred from other experiments, along with the best currently accepted experimental value for ξ 0 , ξ 0 = 1 . 432 A , then the maximum value of the force is predicted to be − 1 . 58 . The effective theory considered here is not consistent with critical point universality; furthermore it incorrectly predicts ordering in the film in violation of known rigorous results. These issues are discussed in the text.