  The Casimir effect is induced by the interplay between photon fields and
boundary conditions, and in particular, photon fields modified in axion
electrodynamics may lead to the sign-flipping of the Casimir energy. We propose
a theoretical approach to derive the Casimir effect in axion electrodynamics.
This approach is based on a lattice regularization and enables us to discuss
the dependence on the lattice spacing for the Casimir energy. With this
approach, the sign-flipping behavior of the Casimir energy is correctly
reproduced. By taking the continuum limit of physical quantity calculated on
the lattice, we can obtain the results consistent with the continuum theory.
This approach can also be applied to the Casimir effect at nonzero temperature.
