  A dissipationless longitudinal current can be generated by a pure magnetic
field through the chiral magnetic effect. Herein, we propose that a pure
oscillating magnetic field through Zeeman coupling can further drive an AC
magnetic Hall current in two-dimensional systems without inversion symmetry. We
dub this effect the "gyrotropic magnetic Hall effect" (GMHE), in analogy with
the gyrotropic current achieved by rectifying the optical fields. Importantly,
we find that the GMHE conductivity is a reactive or dissipationless transport
coefficient, which is even under time-reversal symmetry. We reveal the "Zeeman
Berry curvature" as the quantum origin of the GMHE, whose integral over all
states below the Fermi energy gives the GMHE conductivity. Furthermore, by
symmetry analysis, we show that the GMHE can appear in a wide range of
two-dimensional materials. To demonstrate our proposal, we evaluate the GMHE
current in two-dimensional Rashba system and in the surface of topological
insulator, where a low-frequency magnetic field with a small amplitude can be
converted into a detectable Hall voltage.
