Fermionic Casimir effect for parallel plates in the presence of compact dimensions with applications to nanotubes

TL;DR

This paper calculates the fermionic Casimir energy and force between parallel plates in a spacetime with compact dimensions, with applications to nanotubes, revealing force behaviors depending on nanotube type and length.

Contribution

It provides a general framework for evaluating fermionic Casimir effects in higher-dimensional compactified spaces with boundary conditions, applied to nanotubes.

Findings

Interaction Casimir force is always negative (attractive).

For metallic nanotubes, the Casimir force is always attractive.

For semiconducting nanotubes, the force can switch from attractive to repulsive depending on length.

Abstract

We evaluate the Casimir energy and force for a massive fermionic field in the geometry of two parallel plates on background of Minkowski spacetime with an arbitrary number of toroidally compactified spatial dimensions. The bag boundary conditions are imposed on the plates and periodicity conditions with arbitrary phases are considered along the compact dimensions. The Casimir energy is decomposed into purely topological, single plate and interaction parts. With independence of the lengths of the compact dimensions and the phases in the periodicity conditions, the interaction part of the Casimir energy is always negative. In order to obtain the resulting force, the contributions from both sides of the plates must be taken into account. Then, the forces coming from the topological parts of the vacuum energy cancel out and only the interaction term contributes to the Casimir force. Applications of the general formulae to Kaluza-Klein type models and carbon nanotubes are given. In particular, we show that for finite length metallic nanotubes the Casimir forces acting on the tube edges are always attractive, whereas for semiconducting-type ones they are attractive for small lengths of the nanotube and repulsive for large lengths.