From Dirac surface response to Casimir forces in Bumblebee-CFJ electrodynamics

We examine the Casimir interaction between planar massive Dirac materials coupled to a Lorentz-violating Bumblebee-CFJ vector sector. The standard renormalized one-loop polarization tensor of the Dirac quasiparticles is used as the nonlocal surface response in the Lifshitz scattering formalism. For the normally oriented Lorentz-violating background, the surface response remains parity-even. Hence, the repulsive regimes do not originate from a Hall-like conductivity. Instead, they arise from the CFJ-induced mixing of TE and TM bulk modes, combined with the spatially dispersive response of the Dirac sheets. Furthermore, we show that the Dirac mass suppresses the interaction and changes its large-distance scaling, whereas the CFJ parameter drives transitions between attractive and repulsive regimes.