Itinerant antiferromagnetic BaMn$_2$Pn$_2$s showing both negative and positive magnetoresistances

We report the discovery of a novel giant magnetoresistance (GMR) phenomenon in a family of BaMn$_{2}$Pn$_{2}$ antiferromagnets (Pn stands for P, As, Sb, and Bi) with a parity-time symmetry. The resistivities of these materials are reduced by $60$ times in magnetic fields ($vec{H}$s), thus yielding the GMR of about $-98%$. The GMR changes systematically along with the Pn elements, hinting that its origin is the spin orbit coupling (SOC) and/or $d$-$p$ orbital hybridization. A positive MR component emerging on top of the negative GMR at low temperatures suggests an orbital-sensitive magnetotransport as $vec{H}$ suppresses the conduction of the electron-like carriers in the $d$-like band but enhances those of hole-like ones in the $d$-$p$ hybridized band. The anisotropy of the GMR reveals that the electrical conductivity is extremely sensitive to the minute changes in the direction of the antiferromagnetic moments induced by the parity-time breaking $vec{H}$, which seems to be associated with a magnetoelectric effect in the dynamic regime of conduction electrons. We attribute the observed GMR to the non-trivial low energy band of BMPns, which is governed by the parity-time symmetry and an magnetic hexadecapole ordering.