The Casimir Effect for Lattice Fermions

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 Möbius 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.

Figures (14)

• Figure 1: The link variables $U_\mu(n)$ and $U_{-\mu}(n)$
• Figure 2: The four link variables which build up the plaquette $P_{\mu\nu}(n)$ . The circle indicates the order in which the links are run through in the plaquette.
• Figure 3: Dispersion Relations for massless and massive, naive and Wilson lattice fermions in ($1+1$)- dimensional spacetime.
• Figure 4: Here, $L$ is the transverse dimension of the plate. The distance between the plates is $d$, and the area of the plates $(A) = L^2$.
• Figure 5: (a) Color permeability and permittivity in the bag-model. The outward pressure from the quark fields is balanced by the inward pressure on the spherical cavity modelled as a hadron. (b) Color fields at the surface of the bag.