Thermodynamic Casimir effect: Universality and corrections to scaling

Abstract

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 the hypothesis that corrections are well described by an effective thickness ${L}_{0,\mathrm{eff}}={L}_{0}+c{({L}_{0}+{L}_{s})}^{1\ensuremath{-}\ensuremath{\omega}}+{L}_{s}$, where $c$ and ${L}_{s}$ are system specific parameters and $\ensuremath{\omega}\ensuremath{\approx}0.8$ is the exponent of the leading bulk correction. We simulate the improved Blume-Capel model and the spin-1/2 Ising model on the simple cubic lattice. First, we analyze the behavior of various quantities at the critical point. Taking into account corrections $\ensuremath{\propto}{L}_{0}^{\ensuremath{-}\ensuremath{\omega}}$ 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 ${\ensuremath{\Delta}}_{+\ensuremath{-}}\ensuremath{-}{\ensuremath{\Delta}}_{++}=3.200(5)$, which nicely compares with ${\ensuremath{\Delta}}_{+\ensuremath{-}}\ensuremath{-}{\ensuremath{\Delta}}_{++}=3.208(5)$ obtained by studying the improved Blume-Capel model. Next, we study the behavior of the thermodynamic Casimir force for large values of the scaling variable $x=t{({L}_{0}/{\ensuremath{\xi}}_{0})}^{1/\ensuremath{\nu}}$. It can be obtained up to an overall amplitude by expressing the partition function of the film in terms of eigenvalues and eigenstates of the transfer matrix and boundary states. Here, we demonstrate how this amplitude can be computed with high accuracy. Finally, we discuss our results for the scaling functions ${\ensuremath{\theta}}_{+\ensuremath{-}}$ and ${\ensuremath{\theta}}_{++}$ 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.