  Lifting the valley degeneracy is an efficient way to achieve valley
polarization for further valleytronics operations. In this work, we demonstrate
that a large valley splitting can be obtained in monolayer transition metal
dichalcogenides by magnetic proximity coupling to an insulating
antiferromagnetic substrate. As an example, we perform first-principles
calculations to investigate the electronic structures of monolayer WS$_2$ on
the MnO(111) surface. Our calculation results suggest that a large valley
splitting of 214 meV, which corresponds to a Zeeman magnetic field of 1516 T,
is induced in the valence band of monolayer WS$_2$. The magnitude of valley
splitting relies on the strength of interfacial orbital hybridization, and can
be continually tuned by applying an external out-of-plane pressure and in-plane
strain. More interestingly, we find that both spin and valley index will flip
when the magnetic ordering of MnO is reversed. Besides, owing to the sizeable
Berry curvature and time-reversal symmetry breaking in the WS$_2$/MnO
heterostructure, a spin and valley polarized anomalous Hall current can be
generated in the presence of an in-plane electric field, which allow one to
detect valleys by the electrical approach. Our results shed light on the
realization of valleytronic devices using the antiferromagnetic insulator as
the substrate.
