Revisiting the magnetic responses of bilayer graphene from the perspective of quantum distance

We study the influence of quantum geometry on the magnetic responses of quadratic band crossing semimetals. More explicitly, we examine the Landau levels, quantum Hall effect, and magnetic susceptibility of a general two-band Hamiltonian that has fixed isotropic quadratic band dispersion but with tunable quantum geometry, in which the interband coupling is fully characterized by the maximum quantum distance ${d}_{\mathrm{max}}$. By continuously tuning ${d}_{\mathrm{max}}$ in the range of $0\ensuremath{\le}{d}_{\mathrm{max}}\ensuremath{\le}1$, we investigate how the magnetic properties of the free-electron model with ${d}_{\mathrm{max}}=0$ evolve into those of the bilayer graphene with ${d}_{\mathrm{max}}=1$. We demonstrate that despite sharing the same energy dispersion $\ensuremath{\epsilon}(\mathbit{p})=\ifmmode\pm\else\textpm\fi{}\frac{{p}^{2}}{2m}$, the charge carriers in the free-electron model and bilayer graphene exhibit entirely distinct Landau levels and quantum Hall responses due to the nontrivial quantum geometry of the wave functions.