Surfaces of axion insulators

Axion insulators (AXIs) are magnetic topological insulators in which the non-trivial Ζ₂ index is protected by inversion instead of time-reversal (TR) symmetry. The naturally gapped surfaces of AXIs give rise to a half-quantized surface anomalous Hall conductivity (SAHC), but its sign cannot be determined from topological arguments. To be explicit, we construct a minimal tight-binding model on the pyrochlore lattice and investigate the all-in-all-out and ferromagnetic spin configurations. We show that the Wilson loop eigenvalues are not required to wind for an AXI in contrast with their TR-invariant cousin. We also implement a method that lets us calculate the SAHC directly, which allows us to explore how the interplay between surface termination and magnetic ordering determines the sign of the half-quantized SAHC. We find that it is possible to construct a topological state with no protected metallic states on boundaries of any dimension, although chiral hinge modes do occur for many surface configurations. Finally, we find that rotation of the magnetization by an external field offers promising means of control of chiral hinge modes, which can also appear on surface steps or where bulk domain walls emerge at the surface.