The Casimir Effect for Lattice Fermions

TL;DR

This paper investigates the Casimir effect for lattice fermions, comparing different formulations and boundary conditions, and confirms that naive fermions can reproduce continuum results in the appropriate limit.

Contribution

It provides analytical and numerical analysis of the Casimir effect for various lattice fermion formulations, clarifying their continuum limit behavior and addressing previous misconceptions.

Findings

Lattice results agree with continuum expressions as lattice spacing vanishes.

Naive fermions show fermion doubling effects but can reproduce the Casimir effect in the continuum limit.

Wilson and overlap fermions reproduce expected continuum behavior under different boundary conditions.

Abstract

The Casimir effect for photons and Dirac fermion fields, and its generalization to $(D+1)$-dimensional spacetime in the continuum, is studied. We implement MIT bag boundary conditions on the lattice by treating the system as a confined fermionic slab with perfectly conducting parallel plates. Using the formalism developed for lattice fermions, we compute the Casimir energy for free naive and Wilson fermions analytically in $(1+1)$ dimensions using the Abel-Plana formula, and numerically in higher dimensions. The Casimir energy for overlap fermions with a M\"obius domain wall kernel is also evaluated numerically. For MIT bag boundary conditions, the lattice results agree with the continuum expressions in the limit of vanishing lattice spacing for all fermion formulations. Fermion doubling effects are observed for naive fermions. No oscillatory behavior of the Casimir energy is observed in this case for either massive or massless fermions. We also study periodic and antiperiodic boundary conditions. In these cases, Wilson and overlap fermions reproduce the expected continuum behavior, while naive fermions exhibit oscillations with even and odd lattice sizes and approach different limits. Contrary to earlier claims, our numerical results indicate that naive fermions can reproduce the Casimir effect for Dirac fermions in the appropriate continuum limit, consistent with universality. Finally, we comment on extensions to negative-mass Wilson and overlap fermions and their connection to bulk and surface states in topological insulators.