A general relation between stacking order and Chern index: a topological map of minimally twisted bilayer graphene

TL;DR

This paper establishes a universal relation linking stacking configurations in bilayer graphene to its topological Chern index, revealing that topological properties are robust against lattice relaxations and variations.

Contribution

It derives a general formula connecting stacking vector and Chern index, showing topological invariance despite lattice relaxation in twisted bilayer graphene.

Findings

Chern index depends on stacking vector and valley index

Topological maps are identical for ideal and relaxed structures

Topological properties are robust to lattice relaxations

Abstract

We derive a general relation between the stacking vector ${\bf u}$ describing the relative shift of two layers of bilayer graphene and the Chern index. We find $C = \nu - \text{sign}\left(|V_{AB}|-|V_{BA}|\right)$, where $\nu$ is a valley index and $|V_{\alpha\beta}|$ the absolute value of stacking potentials that depend on ${\bf u}$ and that uniquely determine the interlayer interaction; AA stacking plays no role in the topological character. With this expression we show that while ideal and relaxed minimally twisted bilayer graphene appear so distinct as to be almost different materials, their Chern index maps are, remarkably, identical. The topological physics of this material is thus strongly robust to lattice relaxations.