Casimir effect on the lattice spacetime

We discuss the properties of the Casimir energy on the lattice spacetime. In general, the Casimir energy is defined with a regularization to eliminate the divergence of zero-point energy for a quantum field. Here, we apply the lattice regularization to define the Casimir energy, where lattice effects are implemented as long as the lattice spacing is nonzero. First, we demonstrate the calculation procedure of the Casimir energies using the naive and Wilson fermions. If the lattice effect appears only at a small number of lattices, we can take the continuum limit to obtain the Casimir energy in the continuum spacetime. As examples of physical systems, we investigate the Casimir energy for electron fields in Dirac/Weyl semimetals, where the lattice effect is correctly contained, and for photon fields in axion electrodynamics, where the continuum result is correctly reproduced.