Non-Hermitian tight-binding bands in graphene: Optical conductivity, strain effects, and Bernal bilayer extensions

We introduce a geometry‑dependent tight‑binding model for graphene π‑electrons—the Tan‑Bo model—in which nearest‑neighbour hopping integrals are parametrized by a Möbius angular factor combined with exponential decay. This construction naturally captures bond‑orientation anisotropy while preserving the gapless Dirac cone at the K point under Hermitian assembly. When the reverse‑bond hopping violates the complex‑conjugation relation, the non‑Hermitian assembly yields complex eigenenergies that diagnose non‑reciprocal band structures rather than quasiparticle lifetimes. Benchmarking against the Slater–Koster scaling demonstrates exact equivalence in the isotropic limit. Embedding the Tan‑Bo intralayer fk into the Bernal bilayer McCann model recovers the robust linear gap‑bias relation ΔK≈2δ. Under uniaxial strain, a hybrid scheme combining Harrison/Pereira distance scaling with Möbius angular modulation generates tunable real‑band separation and non‑Hermitian imaginary components. Kubo optical conductivity reproduces the universal Dirac plateau, the TB van Hove structure, and strain‑induced σxx/σyy dichroism qualitatively consistent with polarized transmittance trends. This work provides a reproducible single‑particle benchmark and parametrization reference for future non‑Hermitian calculations including correlation effects.