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Spontaneous symmetry breakings in two-dimensional kagome lattice

We study spontaneous symmetry breakings for fermions (spinless and spinful) on a two-dimensional kagome lattice with nearest-neighbor repulsive interactions in weak coupling limit, and focus in particular on topological Mott insulator instability. It is found that at $\frac{1}{3}$-filling where there is a quadratic band crossing at $Γ$-point, in agreement with Ref. 1, the instabilities are infinitesimal and topological phases are dynamically generated. At $\frac{2}{3}$-filling where there are two inequivalent Dirac points, the instabilities are finite, and no topological phase is favored at this filling without breaking the lattice translational symmetry. A ferromagnetic quantum anomalous Hall state with infinitesimal instability is further proposed at half-filling of the bottom flat band.