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

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 $\mathrm{\mid }\mathbf{p}{\mathrm{\mid }}^{\alpha }$ with $\alpha \ge 2$ treated as a parameter. We derive a closed, analytical expression for the Casimir force depending on the dimensionality $d$ and the exponent $\alpha$, and analyze it for thermodynamic states in which the Bose-Einstein condensate is present. For $\alpha \in \{4,6,8,\dots \}$ the exponent governing the decay of the Casimir force with increasing distance between the walls becomes modified and the Casimir amplitude ${\mathrm{\Delta }}_{\alpha }(d)$ exhibits oscillations of sign as a function of $d$. Otherwise we find that ${\mathrm{\Delta }}_{\alpha }(d)$ features singularities when viewed as a function of $d$ and $\alpha$. Recovering the known previous results for the isotropic limit $\alpha =2$ turns out to occur via a cancellation of singular terms.