Realization of 2D Aubry-André Model in Magic-angle Twisted Bilayer Graphene-hBN Quasicrystal

Quasiperiodicity in crystals corresponds to long range order with broken translational symmetry due to forbidden crystallographic tiling. In this work, we study the nature of the electronic wavefunction of an effective quasiperiodic structure from two overlapping moiré patterns, namely magic angle twisted bilayer graphene (MATBG) and twisted graphene-hBN. We demonstrate that this system realizes a 2D analogue of the Aubry-André-Harper model, where the moiré quasiperiodicity alone generates fractal band structures without an external magnetic field. Using scanning tunneling microscopy and spectroscopy, we show that multifractal critical states emerge specifically within the flat bands. In this regime, the suppressed kinetic energy allows the quasiperiodic potential to dominate, evidenced by a significant increase in the singularity spectrum width ( ) and a reduction in the correlation dimension ( ). Conversely, as we approach the remote bands, the higher kinetic energy overcomes the quasiperiodic potential, restoring non-fractal metallic characteristics. However, real-space imaging reveals that even at these higher energies, the electrons become delocalized along certain directions at specific bias voltages, breaking certain rotational symmetries. These results establish MATBG-hBN as a robust platform for studying the transition from localized AA-site states to multifractal and directionally delocalized phases driven by lattice geometry.