Schematics of the low-energy band structure. An A-type antiferromagnetic bilayer hosts switchable spin-up (-down) bands, located in separate layers. The red (blue) spin-up (-down) bands represent the dispersion relation of states in the top (bottom) layer. At zero displacement field ( D / ε 0 = 0, top panel), the potential difference Δ U between the layers vanishes, and the spin-up and -down bands are degenerate. At finite Δ U (D / ε 0 ≠ 0, middle and bottom panels), the interlayer potential difference lowers the energy of states in one layer (hence with one spin polarity) relative to the other, thereby breaking inversion ( P ) and space-time ( PT ) symmetry. Reversing the sign of D / ε 0 lowers the energy of states located in the other layer (bottom panel), with opposite spin (Supplementary Fig. 1 ).
