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Motivated by recent experiments on AV$_3$Sb$_5$ (A=K,Rb,Cs), the chiral fluxphase has been proposed to explain time-reversal symmetry breaking. To fullyunderstand the physics behind the chiral flux phase, we construct a low-energyeffective theory based on the van-Hove points around the Fermi surface. Thepossible symmetry-breaking states and their classifications of the low-energyeffective theory are completely studied, especially the flux phases on Kagomelattice. In addition, we discuss the relations between the low-energy symmetrybreaking orders, the chiral flux and charge bond orders. We find all possible183 flux phases on Kagome lattice within 2*2 unit cell by brute-force approachand classify them by point group symmetry. Among the 183 phases, we find 3classes in 1*1 unit cell, 8 classes in 1*2 unit cell and 18 classes in 2*2 unitcell, respectively. These results provide a full picture of the time-reversalsymmetry breaking in Kagome lattices.

Low-energy effective theory and symmetry classification of flux phases on Kagome lattice