Specific heat, internal energy, and thermodynamic Casimir force in the neighborhood of the λ transition - Hasenbusch, Martin (Physical review. B, Condensed matter and materials physics, 2010)
We discuss the relation of the excess specific heat, the excess energy per area, and the thermodynamic Casimir force in thin films. A priori these quantities depend on the reduced temperature t and the thickness L0 of the film. However finite-size scaling theory predicts that the scaling functions h′′(x), h′(x), and θ(x) of these quantities depend only on the combination x=t[L0/ξ0]1/ν, where ν is the critical exponent and ξ0 the amplitude of the correlation length. Furthermore, the finite-size scaling function θ(x) of the thermodynamic Casimir force per area can be expressed in terms of the scaling functions h′(x) and h(x) of the excess energy per area and the excess free energy per area. Here we study this relation at the example of thin films of the improved two-component ϕ4 model on the simple cubic lattice. Note that this model undergoes a second-order phase transition that belongs to the three-dimensional XY universality class. First we simulate films with periodic boundary conditions in the short direction and a thickness up to L0=13 lattice spacings. We find that even for these rather thin films, the predictions of finite-size scaling are well satisfied. We repeat the analysis for films with free boundary conditions. To this end we use Monte Carlo data for the energy per area obtained in previous work. It turns our that corrections to scaling caused by the boundary conditions are very prominent in this case. Only by taking into account these corrections we are able to obtain θ(x) from the excess energy. Finally we repeat this exercise using experimental data for the excess specific heat of 4He films near the λ transition. The finite-size scaling behavior of the excess specific heat is governed by h′′(x), which is proportional to the scaling function f2 discussed in the literature.