Dirac Electrons in AC-Magnetic Fields: $\pi$-Landau Levels and Chiral Anomaly-Induced Homodyne Effect

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-\textit{electric} fields as the controlling drive. In this study, we investigate the effects of AC-\textit{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, the 2D Dirac electrons form a new localized state with a flat band dispersion, dubbed as a $\pi$-Landau level. Its wave function is a superposition of the clockwise and counterclockwise cyclotron orbits with time-periodic amplitudes, resulting in a novel closed trajectory shaped like a figure eight. Then, what would be the counterpart of the Hall effect in AC-magnetic fields? We find that a DC-current in the transverse direction, \textit{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.