Bulk quantum Hall effect of spin-valley-coupled Dirac fermions in a polar antiferromagnet BaMnSb$_2$

Unconventional features of relativistic Dirac/Weyl quasi-particles in topological materials are most evidently manifested in the 2D quantum Hall effect (QHE), whose variety is further enriched by their spin and/or valley polarization. Although its extension to three dimensions has been long-sought and inspired theoretical proposals, material candidates have been lacking. Here we have discovered valley-contrasting spin-polarized Dirac fermions in a multilayer form in bulk antiferromagnet BaMnSb$_2$, where the out-of-plane Zeeman-type spin splitting is induced by the in-plane inversion symmetry breaking and spin-orbit coupling (SOC) in the distorted Sb square net. Furthermore, we have observed well-defined quantized Hall plateaus together with vanishing interlayer conductivity at low temperatures as a hallmark of the half-integer QHE in a bulk form. The Hall conductance of each layer is found to be nearly quantized to $2(N+1/2)e^2/h$ with $N$ being the Landau index, which is consistent with two spin-polarized Dirac valleys protected by the strong spin-valley coupling.