Critical free energy and Casimir forces in rectangular geometries

Abstract

We study the critical behavior of the free energy and the thermodynamic Casimir force in a $L_{\parallel }^{d-1}\times L$ block geometry in $2<d<4$ dimensions with aspect ratio $\rho =L/L_{\parallel }$ above, at, and below $T_c$ on the basis of the O$(n)$ symmetric ${\varphi }^4$ lattice model with periodic boundary conditions (b.c.). We consider a simple-cubic lattice with isotropic short-range interactions. Exact results are derived in the large - $n$ limit describing the geometric crossover from film ($\rho =0$) over cubic $\rho =1$ to cylindrical ($\rho =\infty$) geometries. For $n=1$, three perturbation approaches are presented that cover both the central finite-size regime near $T_c$ for $1/4\lesssim \rho \lesssim 3$ and the region outside the central finite-size regime well above and below $T_c$ for arbitrary $\rho$. At bulk $T_c$ of isotropic systems with periodic b.c., we predict the critical Casimir force in the vertical $(L)$ direction to be negative (attractive) for a slab ($\rho <1$), positive (repulsive) for a rod ($\rho >1$), and zero for a cube $(\rho =1)$. We also present extrapolations to the cylinder limit ($\rho =\infty$) and to the film limit ($\rho =0$) for $n=1$ and $d=3$. Our analytic results for finite-size scaling functions in the minimal renormalization scheme at fixed dimension $d=3$ agree well with Monte Carlo data for the three-dimensional Ising model by Hasenbusch for $\rho =1$ and by Vasilyev et al. for $\rho =1/6$ above, at, and below $T_c$.

23 pages, 14 figures

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