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We show that topological states are often developed in two dimensionalsemimetals with quadratic band crossing points (BCPs) by electron-electroninteractions. To illustrate this, we construct a concrete model with the BCP onan extended Lieb lattice and investigate the interaction-driven topologicalinstabilities. We find that the BCP is marginally unstable againstinfinitesimal repulsions. Depending on the interaction strengths, topologicalquantum anomalous/spin Hall, charge nematic, and nematic-spin-nematic phases,develop separately. Possible physical realizations of quadratic BCPs areprovided.

Interaction-driven topological and nematic phases on the Lieb lattice