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 $\rho=L_{\perp}/L_{\parallel}$ 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 [1]. 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 $\rho$ and compared to the three-dimensional case. We give a general argument that the Casimir force vanishes at the critical point for $\rho=1$ and becomes repulsive in periodic systems for $\rho>1$. \\[4pt] [1] Alfred Hucht, Daniel Gr\"uneberg, and Felix M. Schmidt, Phys. Rev. E 83, 051101 (2011)