Twisted bilayer graphene (TBG) is known to have disorder in its twist angle. We show that in terms of a Dirac equation with a random gauge potential $\mathbf{A}\left(\mathbf{r}\right)$, this disorder becomes huge when the average twist angle is near the magic angle where the Dirac velocity vanishes. The density of states then diverges at the Dirac point as $\rho \left(E\right)\sim E^{(2/z)-1}$ with $z\gg 1$, and we deduce that electrons occupy energies very near $E=0$. We solve a strong disorder problem by deriving a sum rule on the disorder averaged eigenfunctions from which we deduce that each added electron contributes equal intraband Coulomb interaction energy. The various bands in TBG are related by either $\mathbf{A}\left(\mathbf{r}\right)\to \mathbf{A}(-\mathbf{r})$ or $\mathbf{A}\left(\mathbf{r}\right)\to -\mathbf{A}\left(\mathbf{r}\right)$, which affects the interband interaction energy. We find, within Hartree-Fock, jumps in the chemical potential at each integer filling, accounting for the cascade transitions.

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• Density of states
• Surface states
• Disordered systems