  Berry curvature multipoles appearing in topological quantum materials have
recently attracted much attention. Their presence can manifest in novel
phenomena, such as nonlinear anomalous Hall effects (NLAHE). The notion of
Berry curvature multipoles extends our understanding of Berry curvature effects
on the material properties. Hence, research on this subject is of fundamental
importance and may also enable future applications in energy harvesting and
high-frequency technology. It was shown that a Berry curvature dipole can give
rise to a 2nd order NLAHE in materials of low crystalline symmetry. Here, we
demonstrate a fundamentally new mechanism for Berry curvature multipoles in
antiferromagnets that are supported by the underlying magnetic symmetries.
Carrying out electric transport measurements on the kagome antiferromagnet
FeSn, we observe a 3rd order NLAHE, which appears as a transverse voltage
response at the 3rd harmonic frequency when a longitudinal a.c. current drive
is applied. Interestingly, this NLAHE is strongest at and above room
temperature. We combine these measurements with a scaling law analysis, a
symmetry analysis, model calculations, first-principle calculations, and
magnetic Monte-Carlo simulations to show that the observed NLAHE is induced by
a Berry curvature quadrupole appearing in the spin-canted state of FeSn. At a
practical level, our study establishes NLAHE as a sensitive probe of
antiferromagnetic phase transitions in other materials, such as moir\'e
superlattices, two-dimensional van der Waal magnets, and quantum spin liquid
candidates, that remain poorly understood to date. More broadly, Berry
curvature multipole effects are predicted to exist for 90 magnetic point
groups. Hence, our work opens a new research area to study a variety of
topological magnetic materials through nonlinear measurement protocols.
