Klein tunneling in quantum geometric semimetals

Klein tunneling stands as a fundamental probe of relativistic quantum transport in two-dimensional materials. We investigate this phenomenon in quadratic band-touching systems, where the Hilbert-Schmidt quantum distance plays a central role in the underlying mechanism. By employing a generic parabolic model, we systematically disentangle the cooperative effects of intrinsic mass asymmetry and tunable quantum geometry. We demonstrate that mass asymmetry sets the overall transmission profile, including the angular distribution and the resonance channels. In contrast, we show that quantum geometry provides a universal parameter that modulates tunneling efficiency by tuning the quantum distance, while leaving the energy dispersion unchanged. Specifically, quantum geometry plays a dual role: it governs the overall transmission amplitude through pseudospin mismatch, while its interplay with Fabry-Perot interference induces observable shifts in resonance angles. Our findings reveal that incorporating quantum geometry alongside band structure is essential for a complete description of quantum transport.