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Aspect-ratio dependence of thermodynamic Casimir forces

We consider the three-dimensional Ising model in a $L_\perp \times L_\parallel \times L_\parallel$ cuboid geometry with finite aspect ratio $ρ= L_\perp/L_\parallel$ and periodic boundary conditions along all directions. For this model the finite-size scaling functions of the excess free energy and thermodynamic Casimir force are evaluated numerically by means of Monte Carlo simulations. The Monte Carlo results compare well with recent field theoretical results for the Ising universality class at temperatures above and slightly below the bulk critical temperature $T_\mathrm{c}$. Furthermore, the excess free energy and Casimir force scaling functions of the two-dimensional Ising model are calculated exactly for arbitrary $ρ$ and compared to the three-dimensional case. We give a general argument that the Casimir force vanishes at the critical point for $ρ= 1$ and becomes repulsive in periodic systems for $ρ> 1$.