Exploring new Hall effect is always a fascinating research topic. The ordinary Hall effect and the quantum Hall effect, initially discovered in two-dimensional (2D) nonmagnetic systems, are the phenomena that a transverse current is generated when a system carrying an electron current is placed in a magnetic field perpendicular to the currents. In this Letter, we propose the electric counterparts of these two Hall effects, termed the “electric Hall effect” (EHE) and the “quantum electric Hall effect” (QEHE). The EHE and QEHE emerge in 2D magnetic systems, where the transverse current is generated by applying an electric gate field instead of a magnetic field. We present a symmetry requirement for both intrinsic EHE and QEHE. Besides, we establish an analytical expression for the intrinsic EHE coefficient when the gate field is weak. We show that it is determined by two band geometric quantities: Berry curvature polarization and Berry curvature polarizability. Via first-principles calculations, we investigate the EHE in the monolayer $\mathrm{Ca}{(\mathrm{FeN})}_2$, where significant EHE coefficient is observed around band crossings. Furthermore, we demonstrate that the QEHE can appear in the semiconductor monolayer $\mathrm{BaM}{\mathrm{n}}_2{\mathrm{S}}_3$, of which the Hall conductivity exhibits steps that take on the quantized values 0 and $\pm 1$ in the unit of $e^2/h$ by varying the gate field within the experimentally achievable range. Because of the great tunability of the electric gate field, the EHE and QEHE proposed here can be easily controlled, and should have significant potential applications.

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• Hall effect
• Transport phenomena
• 2-dimensional systems
• Altermagnets
• Magnetic systems