  We theoretically investigate an intrinsic nonlinear anomalous Hall effect
(INAHE) in space-time ($\mathcal{PT}$) symmetric antiferromagnetic metals. The
INAHE is characterized by an asymmetric and non-dissipative part of the
second-order electric conductivity tensor in the clean limit in contrast to the
Drude-type symmetric conductivity tensor with dissipation. By introducing a
multipole description, we show that the emergence of the INAHE is due to active
odd-parity magnetic quadrupoles or magnetic toroidal dipoles under magnetic
orderings. In order to clarify the microscopic origin of the INAHE, we
specifically consider a fundamental tight-binding model of a three-dimensional
tetragonal system. We demonstrate that the INAHE arises from the effective
coupling between magnetic ordering and antisymmetric spin--orbit interaction.
We also discuss essential electron hopping paths driving the INAHE.
