Boundary conditions and the critical Casimir force on an Ising model film: Exact results in one and two dimensions

Abstract

Finite-size effects in certain critical systems can be understood as universal Casimir forces. Here, we compare the Casimir force for free, fixed, periodic, and antiperiodic boundary conditions in the exactly calculable case of the ferromagnetic Ising model in one and two dimensions. We employ a procedure which allows us to calculate the Casimir force with the aforementioned boundary conditions analytically in a transparent manner. Among other results, we find an attractive Casimir force for the case of periodic boundary conditions and a repulsive Casimir force in the antiperiodic case.