  We study the ac Hall response induced by passage of dc transport current in
two- and three-dimensional metals with gyrotropic point groups -- the
gyrotropic Hall effect -- and consider the phenomenon of current-induced
optical activity in noncentrosymmetric metals as a physical application of our
theory. While the effect is expected to be present in single crystals of any
noncentrosymmetric metal, we expect it to be strongest in enantiomorphic Weyl
semimetals. Using the semiclassical kinetic equation approach we present
several mechanisms underlying the gyrotropic Hall effect. Amongst them, the
intrinsic mechanism is determined by the Berry curvature dipole, while
extrinsic impurity-induced processes are related to skew scattering and side
jump phenomena. In general, the intrinsic and extrinsic contributions can be of
similar magnitude. We discuss the gyrotropic Hall effect for all frequencies of
practical interest, from the DC transport limit, to optical frequencies. We
show that for frequencies that are small compared to relevant band splittings,
the trace of the gyrotropic Hall tensor in three-dimensional materials is
proportional to a topological, quantized Berry charge, and therefore is robust
in gyrotropic Weyl systems. This implies that polycrystals of strongly
gyrotropic Weyl semimetals will demonstrate strong current-induced optical
activity, whereas the response vanishes for polycrystalline ordinary metals.
Therefore, the current-induced optical activity can be considered a valuable
tool in identifying the topological nature of a material.
