Nature of even and odd magic angles in helical twisted trilayer graphene

TL;DR

This paper explores the unique properties of flat bands in helical twisted trilayer graphene, revealing how their degeneracy, Chern numbers, and wavefunctions depend on the parity of magic angles and symmetry considerations.

Contribution

It provides analytical expressions for flat band wavefunctions and clarifies the role of symmetries and angle parity in their structure and topological properties.

Findings

Flat bands occur at discrete magic angles with distinct properties.

Odd and even magic angles exhibit different Chern numbers and degeneracies.

Symmetries like $C_{3z}$ and $C_{2y}T$ are crucial in establishing flat bands.

Abstract

Helical twisted trilayer graphene exhibits zero-energy flat bands with large degeneracy in the chiral limit. The flat bands emerge at a discrete set of magic twist angles and feature properties intrinsically distinct from those realized in twisted bilayer graphene. Their degeneracy and the associated band Chern numbers depend on the parity of the magic angles. Two degenerate flat bands with Chern numbers $C_A=2$ and $C_B=-1$ arise at odd magic angles, whereas even magic angles display four flat bands, with Chern number $C_{A/B}=\pm1$, together with a Dirac cone crossing at zero energy. All bands are sublattice polarized. We demonstrate the structure behind these flat bands and obtain analytical expressions for the wavefunctions in all cases. Each magic angle is identified with the vanishing of a zero-mode wavefunction at high-symmetry position and momentum. The whole analytical structure results from whether the vanishing is linear or quadratic for the, respectively, odd and even magic angle. The $C_{3z}$ and $C_{2y}T$ symmetries are shown to play a key role in establishing the flat bands. In contrast, the particle-hole symmetry is not essential, except from gapping out the crossing Dirac cone at even magic angles.