Casimir forces for the ideal Bose gas in anisotropic optical lattices: the effect of alternating sign upon varying dimensionality

Abstract

We analyze the thermodynamic Casimir effect occurring in a gas of non-interacting bosons confined by two parallel walls with a strongly anisotropic dispersion inherited from an underlying lattice. In the direction perpendicular to the confining walls the standard quadratic dispersion is replaced by the term $|{\bf p}|^α$ with $α\geq 2$ treated as a parameter. We derive a closed, analytical expression for the Casimir force depending on the dimensionality $d$ and the exponent $α$, and analyze it for thermodynamic states in which the Bose-Einstein condensate is present. For $α\in\{4,6,8,\dots\}$ the exponent governing the decay of the Casimir force with increasing distance between the walls becomes modified and the Casimir amplitude $Δ_α(d)$ exhibits oscillations of sign as a function of $d$. Otherwise we find that $Δ_α(d)$ features singularities when viewed as a function of $d$ and $α$. Recovering the known previous results for the isotropic limit $α=2$ turns out to occur via a cancellation of singular terms.