Intertwined charge and spin orders through the exchange-interaction on a square lattice

We study exotic electronic charge orders and magnetic spin orders in a lattice model which encloses conducting electrons coupled with localized magnetic moments via the exchange-like interaction. By means of numerical simulations based on kernel polynomials method combined with Langevin dynamics, we systematically map out phase diagrams versus exchange coupling strength. At the half-filling case n=1/2 (one electron per lattice site on average), a robust antiferromagnetic insulator is formed at low temperature regime. Away from the half-filling, competition between itinerant electrons and local spin moments drives intertwined charge and spin orders. As a concrete example, at the three-quarter filling n=3/4, we identify various spin orders such as incommensurate stripe order and a long-sought double-Q state. Especially, we demonstrate that such double-Q magnetic order leads to a Dirac semi-metal phase on the square lattice, which is identified by the electronic density of states and optical conductivity. Additionally, we also uncover the spin dynamics of magnetic orders via Landau-Lifshitz method. These findings demonstrate that the interplay between itinerant electrons and preformed local spin moments via the exchange interaction is able to produce complex magnetic orders and electronic charge orders.