The Axion Insulator as a Pump of Fragile Topology

The axion insulator (AXI) has long been recognized as the simplest example of a 3D magnetic topological insulator (TI). The most familiar AXI results from magnetically gapping the surface states of a 3D $\mathbb{Z}_{2}$ TI while preserving the bulk gap. Like the 3D TI, it exhibits a quantized magnetoelectric polarizability of $\theta=\pi$, and can be diagnosed from bulk symmetry eigenvalues when inversion symmetric. However, whereas a 3D TI is characterized by bulk Wilson loop winding, 2D surface states, and the pumping of the 2D $\mathbb{Z}_{2}$ TI index, we show that an AXI with a large number of bulk bands displays no Wilson loop winding, exhibits chiral hinge states, and does not pump any previously identified quantity. Crucially, as the AXI exhibits the topological angle $\theta=\pi$, its occupied bands cannot be formed into maximally localized symmetric Wannier functions, despite its absence of Wilson loop winding. In this letter, we revisit the AXI from the perspective of the recently introduced notion of "fragile" topology, and discover that it in fact can be generically expressed as the cyclic pumping of a "trivialized" fragile phase: a 2D inversion-symmetric insulator with no Wilson loop winding which nevertheless carries a nontrivial topological index, the nested Berry phase $\gamma_{2}$. We numerically show that the nontrivial value $\gamma_{2}=\pi$ indicates the presence of anomalous 0D corner charges in a 2D insulator, and therefore, that the chiral pumping of $\gamma_{2}$ in a 3D AXI corresponds to the presence of chiral hinge states. We also briefly generalize our results to time-reversal-symmetric higher-order TIs, and discuss the related appearance of nontrivial $\gamma_{2}$ protected by $C_{2}\times\mathcal{T}$ symmetry in twisted bilayer graphene, and its implications for the presence of 0D corner states.