Abstract
The Casimir effect arises from the zero-point energy of particles in momentum space deformed by the existence of two parallel plates. For degrees of freedom on the lattice, its energy-momentum dispersion is determined so as to keep a periodicity within the Brillouin zone, so that its Casimir effect is modified. We study the properties of Casimir effect for lattice fermions, such as the naive fermion, Wilson fermion, and overlap fermion based on the M\"obius domain-wall fermion formulation, in the $1+1$-, $2+1$-, and $3+1$-dimensional space-time with the periodic or antiperiodic boundary condition. An oscillatory behavior of Casimir energy between odd and even lattice size is induced by the contribution of ultraviolet-momentum (doubler) modes, which realizes in the naive fermion, Wilson fermion in a negative mass, and overlap fermions with a large domain-wall height. Our findings can be experimentally observed in condensed matter systems such as topological insulators and also numerically measured in lattice simulations.
Highlights
The Casimir effect [1,2,3,4] is one of the important physical phenomena especially for microscopic systems with spatial boundary conditions
The original Casimir effect was discussed for the photon field, which is described by quantum electrodynamics (QED), but similar concepts can be extended to any field including scalar, fermion [6,7], and other gauge fields, which have been actively studied
In this paper, based on the formulation given by Ref. [70], we investigated the Casimir energies for the free lattice fermions in the 1 + 1, 2 + 1, and 3 + 1-dimensional spacetime with the one spatial direction compactified by the periodic or antiperiodic boundary condition
Summary
The Casimir effect [1,2,3,4] is one of the important physical phenomena especially for microscopic systems with spatial boundary conditions. Wilson fermions with a negative mass are interesting in the sense that they are closely related to the band structures of topological insulators. The Wilson fermion [62,63], overlap fermion [76,77], and domain-wall fermoin [64,65,66] are typical examples of the fermion formulation without doublers in the continuum limit. IV, we investigate the Wilson fermion with a positive or negative mass The latter corresponds to the Casimir effect for the bulk modes of topological insulators. In Appendices B–E, we give derivations of the Casimir energy for the lattice fermions from the Abel-Plana formulas
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