  One big achievement in modern condensed matter physics is the recognition of
the importance of various band geometric quantities in physical effects. As
prominent examples, Berry curvature and the Berry curvature dipole are
connected to the linear and the second-order Hall effects, respectively. Here,
we show that the Berry connection polarizability (BCP) tensor, as another
intrinsic band geometric quantity, plays a key role in the third-order Hall
effect. Based on the extended semiclassical formalism, we develop a theory for
the third-order charge transport and derive explicit formulas for the
third-order conductivity. Our theory is applied to the two-dimensional (2D)
Dirac model to investigate the essential features of the BCP and the
third-order Hall response. We further demonstrate the combination of our theory
with the first-principles calculations to study a concrete material system, the
monolayer FeSe. Our work establishes a foundation for the study of third-order
transport effects, and reveals the third-order Hall effect as a tool for
characterizing a large class of materials and for probing the BCP in band
structure.
