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(∥)(d-1) × L block geometry in 2<d<4 dimensions with aspect ratio ρ=L/L(∥) on the basis of the O(n) symmetric ϕ4 lattice model with periodic boundary conditions and with isotropic short-range interactions. Exact results are derived in the large-n limit describing the geometric crossover from film (ρ=0) over cubic (ρ=1) to cylindrical (ρ=∞) geometries. For n=1, three perturbation approaches in the minimal renormalization scheme at fixed d are presented that cover both the central finite-size regime near T(c) for 1/4≲ρ≲3 and the region well above and below T(c). At bulk T(c), we predict the critical Casimir force in the vertical (L) direction to be negative (attractive) for a slab (ρ<1), positive (repulsive) for a rod (ρ>1), and zero for a cube (ρ=1). Our results for finite-size scaling functions agree well with Monte Carlo data for the three-dimensional Ising model by Hasenbusch for ρ=1 and by Vasilyev et al. for ρ=1/6 above, at, and below T(c).