Casimir effect for fermions on the lattice

The conventional Casimir effect has been studied in the continuous spacetime, but to elucidate its counterpart in the lattice space is an important subject. Here, we discuss various types of Casimir effects for quantum fields on the lattice. By using a definition of the Casimir energy on the lattice, we show that the Casimir effect for the Wilson fermion is similar to that for the continuous Dirac fermion. We apply our definition to an effective Hamiltonian describing Dirac semimetals, such as Cd$_{3}$As$_{2}$ and Na$_{3}$Bi, and find an oscillatory behavior of the Casimir energy as a function of film thickness of semimetals. We also study contributions from Landau levels under magnetic fields and the Casimir effect for nonrelativistic particle fields on the lattice.