  Floquet engineering, which involves controlling systems through time-periodic
driving, is a method for coherently manipulating quantum materials and
realizing dynamical states with novel functionalities. Most research in
solid-state systems has focused on the use of AC-electric fields as the
controlling drive. In this study, we investigate the effects of AC-magnetic
fields on two-dimensional (2D) Dirac electrons and report the emergence of new
states and new transport phenomena. In a magnetic field that temporarily
changes its direction, an electron nor a hole can sorely complete a cyclotron
orbit. However, their resonant state alternating between electron and hole can
do so; This leads to a new localized state that forms a flat band dubbed as a
$\pi$-Landau level. Then, what would be the counterpart of the Hall effect in
AC-magnetic fields? We find that a DC-current in the transverse direction, i.e.
a homodyne Hall current, is generated when an additional AC-electric field is
applied. In the case of Dirac electrons, several electronic states contribute
to this phenomenon including the $\pi$-Landau level. However, when the chemical
potential $\mu$ is near the Dirac point, the dominant contribution comes from
the low-energy electrons and we numerically find the homodyne Hall current to
behave as $I_y=-\frac{e}{h}\mu$ per valley and spin. We explain this phenomenon
through the high-frequency effective Floquet Hamiltonian which resembles the
chiral Landau level Hamiltonian of three-dimensional Weyl Hamiltonian
exhibiting chiral anomaly. We discuss the experimental feasibility and conclude
that it is possible to realize this new exotic state using techniques such as
THz metamaterial enhancement of magnetic fields.
