Large-$n$ approach to thermodynamic Casimir effects in slabs with free surfaces

Abstract

The classical $n$-vector ${\varphi }^4$ model with $O(n)$ symmetrical Hamiltonian $\mathcal{H}$ is considered in a ${\infty }^2\times L$ slab geometry bounded by a pair of parallel free surface planes at separation $L$. The temperature-dependent scaling functions of the excess free energy and the thermodynamic Casimir force are computed in the large-$n$ limit for temperatures $T$ at, above, and below the bulk critical temperature $T_{\mathrm{c}}$. Their $n=\infty$ limits can be expressed exactly in terms of the eigensystem of a self-consistent one-dimensional Schrödinger equation. This equation is solved by numerical means for two distinct discretized versions of the model: in the first ("model A"), only the coordinate $z$ across the slab is discretized and the integrations over momenta conjugate to the lateral coordinates are regularized dimensionally; in the second ("model B"), a simple cubic lattice with periodic boundary conditions along the lateral directions is used. Renormalization-group ideas are invoked to show that, in addition to corrections to scaling $\propto L^{-1}$, anomalous ones $\propto L^{-1}\ln L$ should occur. They can be considerably decreased by taking an appropriate $g\to \infty$ ($T_{\mathrm{c}}\to \infty$) limit of the ${\varphi }^4$ interaction constant $g$. Depending on the model A or B, they can be absorbed completely or to a large extent in an effective thickness $L_{\mathrm{eff}}=L+\delta L$. Excellent data collapses and consistent high-precision results for both models are obtained. The approach to the low-temperature Goldstone values of the scaling functions is shown to involve logarithmic anomalies. The scaling functions exhibit all qualitative features seen in experiments on the thinning of wetting layers of ${}^4$He and Monte Carlo simulations of $XY$ models, including a pronounced minimum of the Casimir force below $T_{\mathrm{c}}$.

24 pages, 6 figures

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