  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.

Mesoscale and Nanoscale Physics