Instabilities of the Lieb Lattice

The Lieb lattice is a bipartite lattice which has three sites per unit cell that are either doubly or four-fold coordinated. Due to the four-fold roational symmetry about the four-fold coordinated sites, the Lieb lattice has states which are localized on the two-fold coordinated sites and produce a perfectly a flat band. The presence of strong interactions is expected to lead to instabilities of the lattice, as has been established by Lieb's Theorem for the half-filled state. .Assuming that the instabilities are of second-order, the boundaries of the paramagnetic phase of the Lieb lattice have been determined by examining the zero frequency limit of the random phase approximation for the susceptibilities. An analysis of the site-dependent components of the susceptibilities, indicates that the paramagnetic state becomes unstable at an infinitesimally small values of the screened Coulomb interaction, U, for particular band fillings. The predicted phase transition include a transition to a partially compensated ferromagnetic state at half-filling, which is in accordance with Lieb's Theorem. The inferred magnetic state is predicted to be stable by Hartree-Fock total energy calculations and is shown to persist in the large U limit of the model. In addition, the RPA calculations show antiferrmagnetic ordering in which the size of the unit cell is quadrupled..