  We introduce a cluster extension of multipole moments to discuss the
anomalous Hall effect (AHE) in both ferromagnetic (FM) and antiferromagnetic
(AFM) states in a unified framework. We first derive general symmetry
requirements for the AHE in the presence or absence of the spin-orbit coupling,
by considering the symmetry of the Berry curvature in k space. The cluster
multipole (CMP) moments are then defined to quantify the macroscopic
magnetization in non-collinear AFM states, as a natural generalization of the
magnetization in FM states. We identify the macroscopic CMP order which induces
the AHE. The theoretical framework is applied to the non-collinear AFM states
of Mn3Ir, for which an AHE was predicted in a first-principles calculation, and
Mn3Z (Z=Sn, Ge), for which a large AHE was recently discovered experimentally.
We further compare the AHE in Mn3Z and bcc Fe in terms of the CMP. We show that
the AHE in Mn3Z is characterized with the magnetization of a cluster octupole
moment in the same manner as that in bcc Fe characterized with the
magnetization of the dipole moment.
