Self-duality properties and localization centers of the electronic wave functions at high magic angles in twisted bilayer graphene

TL;DR

This paper investigates the self-duality and localization of electronic wave functions at high magic angles in twisted bilayer graphene, revealing their Landau state nature, confinement mechanisms, and gauge field analogies, advancing understanding of flat band phenomena.

Contribution

It uncovers the self-duality of flat band states, explains confinement via symmetrized squared Hamiltonian, and explores the transition from non-Abelian to Abelian gauge fields in TBG.

Findings

Flat band states are coherent Landau states with minimal dispersion.

Confinement arises from symmetrized squared moiré potential and orbital quantization.

Transition from non-Abelian to Abelian gauge fields affects magic angle sequences.

Abstract

Twisted bilayer graphene (TBG) is known for exhibiting highly correlated phases at magic angles due to the emergence of flat bands that enhance electron-electron interactions. In the TBG chiral model, electronic wave function properties depend on a single parameter ($\alpha$), inversely proportional to the relative twist angle between the two graphene layers. In previous studies, as the twist angles approached small values, strong confinement, and convergence to coherent Landau states were observed. This work explores flat-band electronic modes, revealing that flat band states exhibit self-duality; they are coherent Landau states in reciprocal space and exhibit minimal dispersion, with standard deviation $\sigma_k=\sqrt{3\alpha/2\pi}$ as $\alpha$ approaches infinity. Subsequently, by symmetrizing the wave functions and considering the squared TBG Hamiltonian, the strong confinement observed in the $\alpha\rightarrow\infty$ limit is explained. This confinement arises from the combination of the symmetrized squared norm of the moir\'e potential and the quantized orbital motion of electrons, effectively creating a quantum well. The ground state of this well, located at defined spots, corresponds to Landau levels with energy determined by the magic angle. Furthermore, we demonstrate that the problem is physically analogous to an electron attached to a non-Abelian $SU(2)$ gauge field with an underlying $C_3$ symmetry. In regions of strong confinement, the system can be considered as Abelian. This allows to define a magnetic energy in which the important role of the wave function parity and gap closing at non-magic angles is revealed. Finally, we investigate the transition from the original non-Abelian nature to an Abelian state by artificially changing the pseudo-magnetic vector components from an $SU(2)$ to a $U(1)$ field, which alters the sequence of magic angles.