Double-exchange ferromagnetism of fermionic atoms in a $p$-orbital hexagonal lattice

A large class of correlated quantum materials feature strong Hund's coupling. Yet cold-atom quantum simulators have so far focused primarily on single-orbital Fermi-Hubbard systems near a Mott insulator. Here we show that repulsively interacting fermions loaded into the $p$-bands of a hexagonal lattice offer a unique platform to study the interplay of "Hundness" and "Mottness." Our theory predicts that the orbital degrees of freedom, despite geometric frustration, produce a rich phase diagram featuring a competing itinerant ferromagnetic (FM) metal and a spin-1 antiferromagnetic (AFM) insulator, with a surprising first-order transition between them controlled by density near half-filling. Ferromagnetism emerges at low fillings from the flat band and persists to stronger interactions and higher fillings via a double-exchange mechanism, where spins align to avoid Hund-rule penalties at the expense of Dirac-fermion kinetic energy. We further argue that the paramagnetic regime is a correlated "Hund metal." $p$-orbital Fermi gases thus provide an ideal experimental setting to investigate competing exchange mechanisms in multi-orbital systems with coexisting localized and itinerant spins.