  The linear intrinsic anomalous Hall effect (IAHE) and second-order IAHE have
been intensively investigated in time-reversal broken systems. However, as one
of the important members of the nonlinear Hall family, the investigation of
third-order IAHE remains absent due to the lack of an appropriate theoretical
approach, although the third-order extrinsic AHE has been studied within the
framework of first- and second-order semiclassical theory. Herein, we
generalize the semiclassical theory for Bloch electrons under the uniform
electric field up to the third-order using wavepacket method and based on which
we predict that the third-order IAHE can also occur in time-reversal broken
systems. Same as the second-order IAHE, we find the band geometric quantity,
the second-order field-dependent Berry curvature arising from the second-order
field-induced positional shift, plays a pivotal role to observe this effect.
Moreover, with symmetry analysis, we find that the third-order IAHE, as the
leading contribution, is supported by 15 time-reversal broken 3D magnetic point
groups (MPGs), corresponding to a wide class of antiferromagnetic (AFM)
materials. Guided by the symmetry arguments, a two-band model is chosen to
demonstrate the generalized theory. Furthermore, the generalized third-order
semiclassical theory depends only on the properties of Bloch bands, implying
that it can also be employed to explore the IAHE in realistic AFM materials, by
combining with first-principles calculations.
