# Localization Trajectory and Chern-Simons axion coupling for Bilayer Quantum Anomalous Hall Systems

## TL;DR

This study explores how bilayer quantum anomalous Hall systems with high Chern numbers behave under strong disorder, revealing complex localization paths and significant magnetoelectric effects linked to the Chern-Simons axion coupling.

## Contribution

It uncovers the rich localization trajectories and stable intermediate states in bilayer QAH systems, extending understanding beyond single-layer models and standard scaling theories.

## Key findings

- Presence of stable intermediate C=1 state tunable by parameters
- Multiple localization scaling patterns beyond Pruisken picture
- Remarkable peak in Chern-Simons axion coupling during localization

## Abstract

Quantum anomalous Hall (QAH) multilayers provide a platform of topological materials with high Chern numbers. We investigate the localization routes of bilayer QAH systems with Chern number C = 2 under strong disorder, by numerical simulations on their quantum transport properties and the Chern-Simons axion coupling. Compared to the single layer counterpart with C = 2, the localization trajectories present much richer behaviors, for example, the existence of the stable intermediate state with C = 1 can be tuned by model parameters. This state was always unstable in the single layer case. Furthermore, the two parameter scaling trajectories also exhibit multiple patterns, some of which were not captured by the standard Pruisken picture. During the process towards localization, the Chern-Simons axion coupling shows a surprisingly remarkable peak which is even higher and sharper in the large size limit. Therefore the disordered bilayer QAH system can be a good candidate for this nontrivial magnetoelectric coupling mediated by orbital motions.