Casimir effect for fermions on the lattice

TL;DR

This paper investigates the Casimir effect for fermions on a lattice, establishing a definition of Casimir energy, comparing lattice and continuous cases, and exploring implications for Dirac semimetals and magnetic fields.

Contribution

It introduces a lattice-based definition of Casimir energy for fermions and demonstrates its similarity to the continuous case, applying it to Dirac semimetals and magnetic field effects.

Findings

Casimir effect for Wilson fermions resembles that for continuous Dirac fermions.

Oscillatory Casimir energy as a function of film thickness in Dirac semimetals.

Magnetic fields influence the Casimir effect via Landau level contributions.

Abstract

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.