Fermionic Casimir densities in toroidally compactified spacetimes with applications to nanotubes

TL;DR

This paper calculates fermionic Casimir densities in higher-dimensional spacetimes with compactified dimensions, applying the results to understand vacuum energies in metallic and semiconducting nanotubes, revealing how topology influences Casimir energy.

Contribution

It provides new analytical formulas for fermionic vacuum expectation values in complex topologies and applies them specifically to nanotubes, highlighting the impact of boundary conditions on Casimir energies.

Findings

Casimir energy is positive for metallic nanotubes.

Casimir energy is negative for semiconducting nanotubes.

Toroidal compactification alters Casimir energy depending on boundary conditions.

Abstract

Fermionic condensate and the vacuum expectation values of the energy-momentum tensor are investigated for a massive spinor fields in higher-dimensional spacetimes with an arbitrary number of toroidally compactified spatial dimensions. By using the Abel-Plana summation formula and the zeta function technique we present the vacuum expectation values in two different forms. Applications of the general formulae to cylindrical and toroidal carbon nanotubes are given. We show that the topological Casimir energy is positive for metallic cylindrical nanotubes and is negative for semiconducting ones. The toroidal compactification of a cylindrical nanotube along its axis increases the Casimir energy for metallic-type (periodic) boundary conditions along its axis and decreases the Casimir energy for the semiconducting-type compactifications.