  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) non-magnetic 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 work, we propose the
electric counterparts of these two Hall effects, termed as electric Hall effect
(EHE) and 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 intrinsic EHE and QEHE. With a weak gate-field, we establish an
analytical expression of the intrinsic EHE coefficient. We show that it is
determined by intrinsic band geometric quantities: Berry curvature and its
polarizability which consists of both intraband and interband layer
polarization. Via first-principles calculations, we investigate the EHE in the
monolayer Ca(FeN)$_2$, where significant EHE coefficient is observed around
band crossings. Furthermore, we demonstrate that the QEHE can appear in the
semiconductor monolayer $\rm BaMn_2S_3$, of which the Hall conductivity
exhibits steps that take on the quantized values $0$ and $\pm1$ in the unit of
$e^2/h$ by varying the gate-field within the experimentally achievable range.
Due to the great tunability of the electric gate-field, the EHE and QEHE
proposed here can be easily controlled and should have more potential
applications.
