  Some antiferromagnets under a magnetic field develop magnetization
perpendicular to the field as well as more conventional ones parallel to the
field. So far, the transverse magnetization (TM) has been attributed to either
spin canting effect or the presence of cluster magnetic multipolar ordering.
However, a general theory of TM based on microscopic understanding is still
missing. Here, we construct a general microscopic theory of TM in
antiferromagnets with cluster magnetic multipolar ordering by considering
classical spin Hamiltonians with spin anisotropy that arises from the
spin-orbit coupling. First, from general symmetry analysis, we show that TM can
appear only when all crystalline symmetries are broken other than the
antiunitary mirror, antiunitary two-fold rotation, and inversion symmetries.
Moreover, by analyzing spin Hamiltonians, we show that TM always appears when
the degenerate ground state manifold of the spin Hamiltonian is discrete. On
the other hand, when the degenerate ground state manifold is continuous, TM
generally does not appear except when the magnetic field direction and the spin
configuration satisfy specific geometric conditions under single-ion
anisotropy. Finally, we show that TM can induce anomalous planar Hall Effect, a
unique transport phenomenon that can be used to probe multipolar
antiferromagnetic structures. We believe that our theory provides a useful
guideline for understanding the anomalous magnetic responses of the
antiferromagnets with complex magnetic structures.
