Effective binding potential from Casimir interactions: the case of the Bose gas

TL;DR

This paper studies the thermal Casimir effect in ideal Bose gases with quadratic and quartic dispersion, revealing how the force's sign depends on separation and dimensionality, leading to an effective binding potential.

Contribution

It demonstrates the dimensional dependence of Casimir energy in Bose gases and derives a simple form at the critical temperature in three dimensions.

Findings

Casimir force changes sign with separation in certain dimensions

Casimir energy is a polynomial in inverse squared separation for odd dimensions

Special behavior of Casimir energy at three dimensions and Bose-Einstein condensation

Abstract

We consider the thermal Casimir effect in ideal Bose gases, where the dispersion relation involves both terms quadratic and quartic in momentum. We demonstrate that if macroscopic objects are immersed in such a fluid in spatial dimensionality $d\in\{3,7, 11, \dots\}$ and at the critical temperature $T_c$, the Casimir force acting between them is characterized by a sign which depends on the separation $D$ between the bodies and changes from attractive at large distances to repulsive at smaller separations. In consequence, an effective potential which binds the two objects at a finite separation arises. We demonstrate that for odd integer dimensionality $d\in \{3, 5, 7, \dots\}$, the Casimir energy is a polynomial of degree $(d-1)$ in $D^{-2}$. We point out a very special role of dimensionality $d=3$, where we derive a strikingly simple form of the Casimir energy as a function of $D$ at Bose-Einstein condensation. We discuss crossover between monotonous and oscillatory decay of the Casimir interaction above the condensation temperature.