Remnants of the nonrelativistic Casimir effect on the lattice

The Casimir effect is a fundamental quantum phenomenon induced by the zero-point energy for a quantum field. It is well-known for relativistic fields with a linear dispersion relation, while its existence or absence for nonrelativistic fields with a quadratic dispersion is an unsettled question. Here, we investigate the Casimir effects for various dispersion relations on the lattice. We find that Casimir effects for dispersions proportional to an even power of momentum are absent in a long distance but a remnant of the Casimir effect survives in a short distance. Such a remnant Casimir effect will be experimentally observed in materials with quantum fields on the lattice, such as thin films, narrow nanoribbons, and short nanowires. In terms of this effect, we also give a reinterpretation of the Casimir effect for massive fields.