Layer skyrmions for ideal Chern bands and twisted bilayer graphene

TL;DR

This paper uncovers how layer skyrmions in ideal Chern bands and twisted bilayer graphene create topologically robust textures that compensate magnetic phases, revealing new gauge symmetries and non-Abelian connections.

Contribution

It demonstrates the emergence of Skyrme textures in Dirac moiré models and their role in ideal Chern bands with higher Chern numbers, linking real-space topology to gauge symmetries.

Findings

Skyrme textures develop in spinor wavefunctions of ideal Chern bands.

A SU(C) gauge symmetry and non-Abelian connection are identified in the layer spinor space.

Skyrme textures remain stable in twisted bilayer graphene under realistic conditions.

Abstract

Ideal $C=1$ Chern bands exhibit a Landau level correspondence: they factorize as a lowest Landau levels and a spinor wavefunction that spans the layer index. We demonstrate that, in single Dirac moir\'e models, the spinor develops generally a Skyrme texture in real space with an associated Berry phase which compensates exactly the magnetic phase of the Landau level. For ideal bands with higher Chern numbers $C>1$, we find that $C$ color Landau levels are carried by $C$ spinors with Skyrme textures. We identify a SU(C) gauge symmetry in the color space of spinors and an emergent non-Abelian connection in real space intimately linked to the Pontryagin winding index of the layer skyrmions. They result in a total real-space Chern number of $-1$, screening the magnetic phase, irrespective of $C$ and of the number of layers. The topologically robust Skyrme texture remains remarkably intact in twisted bilayer graphene, even far from the chiral limit, and for realistic values of corrugation, making it an experimentally testable feature. We verify our predictions at the first magic angle of twisted bilayer, trilayer, and monolayer-bilayer graphene.