The thermodynamic Casimir effect in the neighbourhood of the λ-transition: a Monte Carlo study of an improved three-dimensional lattice model

We study the thermodynamic Casimir effect in thin films in the three-dimensional XY universality class. To this end, we simulate the improved two-component phi⁴ model on the simple cubic lattice. We use lattices up to the thickness L₀ = 33. On the basis of the results of our Monte Carlo simulations we compute the universal finite size scaling function θ that characterizes the behaviour of the thermodynamic Casimir force in the neighbourhood of the critical point. We confirm that leading corrections to the universal finite size scaling behaviour due to free boundary conditions can be expressed using an effective thickness L_0,eff = L₀+L_s, with L_s = 1.02(7). Our results are compared with experiments on films of ⁴He near the λ-transition, previous Monte Carlo simulations of the XY model on the simple cubic lattice and field-theoretic results. Our result for the finite size scaling function θ is essentially consistent with the experiments on films of ⁴He and the previous Monte Carlo simulations.