Casimir effect in a one-dimensional gas of free fermions

TL;DR

This paper investigates the Casimir effect in a one-dimensional free fermion gas, revealing oscillatory interactions and force discontinuities, and confirms these findings through exact solutions and comparisons with effective low-energy theories.

Contribution

It provides a rigorous analysis of the Casimir effect in a 1D fermion gas, resolving previous objections and validating effective low-energy approximations.

Findings

Casimir interaction exhibits oscillatory behavior with force discontinuities.

Maxima of the Casimir force correspond to resonant tunneling.

Regularization results are recovered as the lower envelope of oscillations.

Abstract

We compute an analog Casimir effect in a one-dimensional spinless Luttinger liquid confined to a segment in the presence of a nearly-impenetrable partition dividing the segment into two compartments. The Casimir interaction is found to be a bounded piecewise-continuous oscillatory function whose maxima are points of force discontinuity and correspond to resonant tunneling across the partition. The well-known regularization-based results are reproduced by the lower envelope of this function, which corresponds to an approximation that ignores the rather large oscillations due to particle discreteness. These macroscopic conclusions are tested and confirmed via a rigorous analysis of the Casimir effect in an exactly-solvable model of a one-dimensional non-relativistic spinless gas of free fermions, thus resolving an objection that has been raised by Volovik (2003). Additionally we confirm the result of a recent calculation which employed an effective low-energy theory with a cutoff to find the Casimir interaction between two strong well-separated impurities placed in a Luttinger liquid.