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

TL;DR

This paper provides exact analytical results for the critical Casimir force in the Ising model in one and two dimensions, revealing how boundary conditions influence the force's nature.

Contribution

It introduces a new analytical method to calculate the Casimir force for different boundary conditions in the Ising model, including periodic and antiperiodic cases.

Findings

Periodic boundary conditions lead to an attractive Casimir force.

Antiperiodic boundary conditions result in a repulsive Casimir force.

The method offers transparent calculations in exactly solvable models.

Abstract

Finite size effect in critical systems can be interpreted as universal Casimir forces. Here we compare the Casimir force for free, periodic and antiperiodic boundary conditions in the exactly calculable case of the Ising model in one and two dimensions. We employ a new procedure which allows us to calculate the Casimir force with periodic and antiperiodic boundary conditions analytically in a transparent manner. Quite interestingly, we find an attractive Casimir force for the case of periodic boundary condition but a repulsive Casimir force for the anti-periodic one.