# Topological proximity effects in a Haldane-graphene bilayer system

## TL;DR

This paper investigates the topological proximity effects in a Haldane-graphene bilayer system, revealing how coupling induces topological phases, modifies Berry curvature, and results in edge modes and flat bands, with potential for new topological insulators.

## Contribution

It demonstrates the induction of topological phases and edge states in a Haldane-graphene bilayer through weak coupling, and generalizes the effect to asymmetric systems with new topological invariant definitions.

## Key findings

- Coupling induces a gap and opposite Chern number on Dirac particles.
- Observation protocols for Berry curvature evolution and edge modes are proposed.
- A topological insulator phase with Chern number one is identified when only the lowest band is filled.

## Abstract

We reveal a proximity effect between a topological band (Chern) insulator described by a Haldane model and spin-polarized Dirac particles of a graphene layer. Coupling weakly the two systems through a tunneling term in the bulk, the topological Chern insulator induces a gap and an opposite Chern number on the Dirac particles at half-filling resulting in a sign flip of the Berry curvature at one Dirac point. We study different aspects of the bulk-edge correspondence and present protocols to observe the evolution of the Berry curvature as well as two counter-propagating (protected) edge modes with different velocities. In the strong-coupling limit, the energy spectrum shows flat bands. Therefore we build a perturbation theory and address further the bulk-edge correspondence. We also show the occurrence of a topological insulating phase with Chern number one when only the lowest band is filled. We generalize the effect to Haldane bilayer systems with asymmetric Semenoff masses. We propose an alternative definition of the topological invariant on the Bloch sphere.