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

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.