# Quasicrystalline electronic states in 30$^\circ$ rotated twisted bilayer graphene

## TL;DR

This paper develops a theoretical framework to describe the electronic states in 30° twisted bilayer graphene, revealing quasicrystalline properties, resonant states, and flat bands with fractal wave function patterns.

## Contribution

It introduces a symmetry-respecting theoretical model for 30° twisted bilayer graphene, capturing its quasicrystalline electronic states and resonant phenomena.

## Key findings

- Identification of 12-fold quantized angular momentum states
- Presence of nearly flat bands with fractal wave functions
- Spatial localization of resonant states in finite samples

## Abstract

The recently realized bilayer graphene system with a twist angle of $30^\circ$ offers a new type of quasicrystal which unites the dodecagonal quasicrystalline nature and graphene's relativistic properties. Here, we introduce a concise theoretical framework that fully respects both the dodecagonal rotational symmetry and the massless Dirac nature, to describe the electronic states of the system. We find that the electronic spectrum consists of resonant states labeled by 12-fold quantized angular momentum, together with the extended relativistic states. The resulting quasi-band structure is composed of the nearly flat bands with spiky peaks in the density of states, where the wave functions exhibit characteristic patterns which fit to the fractal inflations of the quasicrystal tiling. We also demonstrate that the 12-fold resonant states appear as spatially-localized states in a finite-size geometry, which is another hallmark of quasicrystal. The theoretical method introduced here is applicable to a broad class of "extrinsic quasicrystals" composed of a pair of two-dimensional crystals overlaid on top of the other with incommensurate configurations.