The Axion Insulator as a Pump of Fragile Topology

Understand

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.