  The topological surface states of magnetic topological systems, such as Weyl
semimetals and axion insulators, are associated with unconventional transport
properties such as nonzero or half-quantized surface anomalous Hall effect.
Here we study the surface anomalous Hall effect and its spectral signatures in
different magnetic topological phases using both model Hamiltonian and
first-principles calculations. We demonstrate that by tailoring the
magnetization and interlayer electron hopping, a rich three-dimensional
topological phase diagram can be established, including three types of
topologically distinct insulating phases bridged by Weyl semimetals, and can be
directly mapped to realistic materials such as MnBi2Te4/(Bi2Te3)n systems.
Among them, we find that the surface anomalous Hall conductivity in the
axion-insulator phase is a well-localized quantity either saturated at or
oscillating around e2/2h, depending on the magnetic homogeneity. We also
discuss the resultant chiral hinge modes embedded inside the side surface bands
as the potential experimental signatures for transport measurements. Our study
is a significant step forward towards the direct realization of long-sought
axion insulators in realistic material systems.
