  Conventional electronics are based invariably on the intrinsic degrees of
freedom of an electron, namely, its charge and spin. The exploration of novel
electronic degrees of freedom has important implications in both basic quantum
physics and advanced information technology. Valley as a new electronic degree
of freedom has received considerable attention in recent years. In this paper,
we develop the theory of spin and valley physics of an antiferromagnetic
honeycomb lattice. We show that by coupling the valley degree of freedom to
antiferromagnetic order, there is an emergent electronic degree of freedom
characterized by the product of spin and valley indices, which leads to
spin-valley dependent optical selection rule and Berry curvature-induced
topological quantum transport. These properties will enable optical
polarization in the spin-valley space, and electrical detection/manipulation
through the induced spin, valley and charge fluxes. The domain walls of an
antiferromagnetic honeycomb lattice harbors valley-protected edge states that
support spin-dependent transport. Finally, we employ first principles
calculations to show that the proposed optoelectronic properties can be
realized in antiferromagnetic manganese chalcogenophosphates (MnPX_3, X = S,
Se) in monolayer form.
