Casimir effect for lattice fermions

TL;DR

This paper defines and analyzes the Casimir energy for various lattice fermions in 1+1 dimensions, revealing behaviors like oscillations and agreement with continuum results, relevant for condensed matter and lattice simulations.

Contribution

It introduces a new definition of Casimir energy for lattice fermions and studies its properties across different fermion formulations in 1+1 dimensions.

Findings

Naive fermions show oscillatory Casimir energy due to lattice size parity.

Wilson fermions' Casimir energy matches continuum results at small lattice sizes.

Parameter tuning in M"obius domain-wall fermions affects the Casimir energy.

Abstract

We propose a definition of the Casimir energy for free lattice fermions. From this definition, we study the Casimir effects for the massless or massive naive fermion, Wilson fermion, and (M\"obius) domain-wall fermion in $1+1$ dimensional spacetime with the spatial periodic or antiperiodic boundary condition. For the naive fermion, we find an oscillatory behavior of the Casimir energy, which is caused by the difference between odd and even lattice sizes. For the Wilson fermion, in the small lattice size of $N \geq 3$, the Casimir energy agrees very well with that of the continuum theory, which suggests that we can control the discretization artifacts for the Casimir effect measured in lattice simulations. We also investigate the dependence on the parameters tunable in M\"obius domain-wall fermions. Our findings will be observed both in condensed matter systems and in lattice simulations with a small size.