Distance-Dependent Sign Reversal in the Casimir-Lifshitz Torque

The Casimir-Lifshitz torque between two biaxially polarizable anisotropic planar slabs is shown to exhibit a nontrivial sign reversal in its rotational sense. The critical distance a_{c} between the slabs that marks this reversal is characterized by the frequency ω_{c}∼c/2a_{c} at which the in-planar polarizabilities along the two principal axes are equal. The two materials seek to align their principal axes of polarizabilities in one direction below a_{c}, while above a_{c} their axes try to align rotated perpendicular relative to their previous minimum energy orientation. The sign reversal disappears in the nonretarded limit. Our perturbative result, derived for the case when the differences in the relative polarizabilities are small, matches excellently with the exact theory for uniaxial materials. We illustrate our results for black phosphorus and phosphorene.