  The control of spin-dependent properties by voltage, not involving
magnetization switching, has significant advantages for low-power spintronics.
Here, we predict that the interfacial crystal Hall effect (ICHE) can serve for
this purpose. We show that the ICHE can occur in heterostructures composed of
compensated antiferromagnetic metals and non-magnetic insulators due to reduced
symmetry at the interface, and it can be made reversible if the antiferromagnet
is layered symmetrically between two identical ferroelectric layers. We
explicitly demonstrate this phenomenon using density functional theory
calculations for three material systems: MnBi$_{2}$Te$_{4}$/GeI$_{2}$ and
topological In$_{2}$Te$_{3}$/MnBi$_{2}$Te$_{4}$/In$_{2}$Te$_{3}$ van der Waals
heterostructures, and GeTe/Ru$_{2}$MnGe/GeTe heterostructure composed of
three-dimensional materials. We show that all three systems reveal a sizable
ICHE, while the latter two exhibit a quantum ICHE and ICHE, respectively,
reversible with ferroelectric polarization. Our proposal opens an alternative
direction for voltage controlled spintronics and offers not yet explored
possibilities for functional devices by heterostructure design.
