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

TL;DR

This paper derives an analytical expression for the Casimir force in an anisotropic Bose gas confined between parallel walls, revealing oscillations in force sign and singularities depending on the dispersion exponent and dimensionality.

Contribution

It introduces a generalized model with variable dispersion exponent, providing new insights into the Casimir effect's dependence on anisotropy and dimensionality in Bose gases.

Findings

Casimir force depends on anisotropic dispersion and dimensionality.

Oscillations in Casimir force sign occur for even integer exponents.

Singularities appear in the Casimir amplitude as a function of parameters.

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 $|{\bf p}|^{\alpha}$ with $\alpha \geq 2$ 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 $\Delta_{\alpha}(d)$ exhibits oscillations of sign as a function of $d$. Otherwise we find that $\Delta_{\alpha}(d)$ 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.