3/2 magic angle quantization rule of flat bands in twisted bilayer graphene and its relationship to the quantum Hall effect

Flat band electronic modes in twisted graphene bilayers are responsible for superconducting and other highly correlated electron-electron phases. Although some hints were known of a possible connection between the quantum Hall effect zero flat modes, it was not clear how such appears. Here behavior bilayer is studied using chiral model Hamiltonian. As result, proved that high-order magic angles, converge into coherent Landau states with dispersion $\sigma^2=1/3\alpha$, where $\alpha$ coupling parameter incorporates twist angle energetic scales. Then square hamiltonian, which $2\times 2$ matrix operator, turns out to be equivalent two-dimensional harmonic oscillator. The interlayer currents graphene's bipartite lattices identified angular momentum term while confinement potential an effective quadratic potential. From there limiting quantization rule i.e., $\alpha_{m+1}-\alpha_{m}=3/2$ $m$ order angle. All these results very good agreement numerical calculations.