Entanglement (1+2) QED in a double layer of Dirac Materials

We investigate the momentum-space entanglement between two Dirac quasiparticles in a double-layer honeycomb lattice coupled via a planar electromagnetic cavity. We model the low-energy excitations as massive Dirac fermions in $(1+2)$ dimensions and derive the Bethe-Salpeter equation using the ladder approximation. We use a Born-level approximation around a free two-body quasiparticle state, where the interaction is mediated by the cavity photon propagator. From the reduced sublattice density matrix, we compute a momentum-resolved von Neumann entropy. Within the perturbatively controlled regime, the entropy remains small, while phenomenological self-energy dressing drives a crossover to strong enhancement of the entanglement entropy. Stationary entanglement is obtained only when the quasiparticle coherence time exceeds the photon propagation time between the layers. The maximum-entropy regime appears to be a viable method for achieving Bell-like states. These results demonstrate how self-energy renormalization, virtual particle exchange, and spinor geometry combine to reshape the entanglement landscape of Dirac materials.