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

The classical $n$-vector ${\varphi }^4$ model with $O\left(n\right)$ 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 $L$. Standard quadratic boundary terms implying Robin boundary conditions are included in $\mathcal{H}$. 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 spectrum and eigenfunctions 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 $\propto L^{-1}lnL$ 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 $T_{\mathrm{c}}$. The results are in conformity with various analytically known exact properties of the scaling functions.

Figure 1

Effective amplitude ${\tilde{\Delta }}_{\mathrm{C}}(g,L)$ for different $g=1,2,4,\cdots ,512,\infty$. Note the strong corrections present at small $g$.

Figure 2

Crossover in $V_z$ at criticality for $L=128$ and different values of $g=1,2,4,\cdots ,128$.

Figure 3

Scaled potential $v\left(\zeta \right)$, Eq. (7.9), as a function of $\zeta =z_{\mathrm{eff}}/L_{\mathrm{eff}}$ at criticality, calculated with model A at $g=\infty$ and different values of $L=2^4,2^5,\cdots ,2^{12}$. The dotted line is an extrapolation to $L\to \infty$ (see text).

Figure 4

Casimir force scaling function $\vartheta \left(x\right)$ (left) and residual free energy scaling function $\Theta \left(x\right)$ (right) determined from data for $g\to \infty$ and $L=65,97,129,193,257$ (model A) and $L=97,129,193,257$ (model B). The dotted curves represent the asymptotic $x\to -\infty$ forms (8.6). For further explanations, see main text. The data for the scaling functions are included in the Supplemental Material [81].

Figure 5

First and second derivative ${\vartheta }^{\prime }\left(x\right)$ (blue solid line) and ${\vartheta }^{\prime \prime }\left(x\right)$ (red dashed line), determined from data for $g\to \infty$ and $L=257$. The horizontal dashed line indicates the exact value ${\vartheta }^{\prime \prime }\left(0\right)=-{\pi }^{-3}$ of Eq. (6.11).

Figure 6

Asymptotic behavior of $\vartheta \left(x\right)$ for $x\to -\infty$. The data are for model A, with $L=9$, 17, 33, 65, 129, 193, 257, 385, 513, 769, 1025, 1537, and are plotted down to $x=-L$. The dashed line is a fit based on Eq. (8.6b) to the data with $m=2$, $c_1=2.0\left(1\right)$, $d_1=1.0\left(1\right)$, $c_2=-17\left(2\right)$, and $d_2=16\left(2\right)$. The dotted line with the slope $-3c_1/2=-3$ is a guide to the eyes (see text).