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Thermodynamic Casimir effect: Universality and Corrections to Scaling

We study the thermodynamic Casimir force for films in the three-dimensional Ising universality class with symmetry breaking boundary conditions. We focus on the effect of corrections to scaling and probe numerically the universality of our results. In particular we check our hypothesis that corrections are well described by an effective thickness L_{0,eff}=L_0+c (L_0+L_s)^{1-ω} +L_s, where c and L_s are system specific parameters and ω\approx 0.8 is the exponent of the leading bulk correction. We simulate the improved Blume-Capel model and the Ising model on the simple cubic lattice. First we analyse the behaviour of various quantities at the critical point. Taking into account corrections \propto L_0^{-ω} in the case of the Ising model, we find good consistency of results obtained from these two different models. In particular we get from the analysis of our data for the Ising model for the difference of Casimir amplitudes Δ_{+-}-Δ_{++}=3.200(5), which nicely compares with Δ_{+-}-Δ_{++}=3.208(5) obtained by studying the improved Blume-Capel model. Next we study the behaviour of the thermodynamic Casimir force for large values of the scaling variable x=t [L_0/ξ_0]. This behaviour can be obtained up to an overall amplitude by expressing the partition function of the film in terms of eigenvalues and eigenstates of the transfermatrix and boundary states. Here we show how this overall amplitude can be computed with high accuracy. Finally we discuss our results for the scaling functions θ_{+-} and θ_{++} of the thermodynamic Casimir force for the whole range of the scaling variable. We conclude that our numerical results are in accordance with universality. Corrections to scaling are well approximated by an effective thickness.