Why the first magic-angle is different from others in twisted graphene bilayers: interlayer currents, kinetic and confinement energy and wavefunction localization

TL;DR

This paper analyzes the physical mechanisms behind the unique properties of the first magic angle in twisted graphene bilayers, focusing on interlayer currents, energies, and wavefunction localization, revealing why it differs from higher angles.

Contribution

It introduces a detailed analysis of the squared chiral Hamiltonian, highlighting the role of interlayer currents and energy contributions in determining magic angles, especially the uniqueness of the first.

Findings

Interlayer currents are linked to non-Abelian interlayer operators.

The first magic angle is distinguished by a specific energy balance.

A perturbative approach clarifies the energy contributions at the first magic angle.

Abstract

The chiral Hamiltonian for twisted graphene bilayers is analyzed in terms of its squared Hamiltonian which removes the particle-hole symmetry and thus one bipartite lattice, allowing to write the Hamiltonian in terms of a $2\times 2$ matrix. This brings to the front the three main physical actors of twisted systems: kinetic energy, confinement potential and an interlayer interaction operator which is divided in two parts: a non-Abelian interlayer operator and an operator which contains an interaction energy between layers. Here, each of these components is analyzed as a function of the angle of rotation, as well as in terms of the wave-function localization properties. In particular, it is proved that the non-Abelian operator represents interlayer currents between each layer triangular sublattices, i.e., a second-neighbor interlayer current between bipartite sublattices. A crossover is seen between such contributions and thus the first magic angle is different from other higher order magic angles. Such angles are determined by a balance between the negative energy contribution from interlayer currents and the positive contributions from the kinetic and confinement energies. A perturbative analysis performed around the first magic angle allows to explore analytically the details of such energy balance.