Magnetic and nematic order of Bose-Fermi mixtures in moiré superlattices of two-dimensional semiconductors

We investigate the magnetic orders in a mixture of a boson (exciton) and a fermion (electron or hole) trapped in transition-metal dichalcogenides moir\'e superlattices. A sizable antiferromagnetic exchange interaction is found between a carrier and an interlayer exciton trapped at different high-symmetry points of the moir\'e supercell. This interaction, at a distance much shorter than the carrier-carrier separation, dominates the magnetic order in the Bose-Fermi mixture, where the carrier sublattice develops ferromagnetism opposite to that in the exciton sublattice. We demonstrate the possibility of increasing the Curie temperature of moir\'e carriers through electrical tuning of the exciton density in the ground state. In a trilayer moir\'e system with a $p\text{\ensuremath{-}}n\text{\ensuremath{-}}p$ type band alignment, the exciton-carrier interplay can establish a layered antiferromagnetism for holes confined in the two outer layers. We further reveal a spontaneous nematic order in the Bose-Fermi mixture, arising from the interference between the Coulomb interaction and $p$-wave interlayer tunneling dictated by the stacking registry.