Nematic Ferromagnetism on the Lieb Lattice

We discuss the properties of ferromagnetic orders on the Lieb lattice and show that a symmetry protected quadratic-flat band crossing point will dramatically affect the magnetic ordering. In the presence of a weak on-site repulsive interaction, the ground state is a nematic ferromagnetic order with simultaneously broken time-reversal and rotational symmetries. When the interaction strength increases, the rotational symmetry will restore at a critical value, and the system enters a conventional ferromagnetic regime. The mean-field transition temperatures for both the nematic and conventional ferromagnetic phases are in the order of interaction. This observation suggests that these magnetic orders have the potential to be realized and detected in cold atomic systems within realistic experimental conditions.