Interaction-driven topological and nematic phases on the Lieb lattice

We investigate the interaction-driven instabilities of the band crossing point (BCP), which occurs in the band structure resulting from spinless/spinful fermions moving on the (extended) Lieb lattice. In the non-interacting limit, we show the topological stability of the BCP both from momentum and real space arguments, provided time reversal and $C_4$ point group symmetries are preserved. With short-range repulsive interactions, we find that at zero temperature this BCP is marginally unstable against infinitesimal repulsions and results in topological quantum anomalous/spin Hall, charge nematic, and nematic-spin-nematic phases, separately, depending on the interaction strengths. Possible physical realizations and the existence of a topological nearly flat band are also discussed.