  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.

Mesoscale and Nanoscale Physics