Complex charge density waves at Van Hove singularity on hexagonal lattices: Haldane-model phase diagram and potential realization in the kagome metals AV3Sb5 ( A =K, Rb, Cs)

We analyze how the real and imaginary charge density waves interplay at the Van Hove singularity on the hexagonal lattices. A phenomenological Ginzburg-Landau analysis indicates the formation of complex orders at all three momenta under a total phase condition. These complex orders break the rotation symmetry universally, since different complex phases are generally carried at the three momenta. A rich Haldane-model phase diagram of 3Q complex charge density waves is mapped out, where the trivial and Chern insulator phases are manifest. These phases are deformations of the purely real and imaginary orders, which exhibit trivial site and/or bond density and chiral flux orders, respectively. The gapless phase boundary may host a Dirac semimetal and an exotic single-Dirac-point semimetal. We further show that the theoretical model offers transparent interpretations of experimental observations in the kagome metals AV3Sb5 with A = K,Rb,Cs. The topological charge density waves may be identified with the complex orders in the Chern insulator phase. Meanwhile, the lower-temperature symmetry-breaking phenomena may be interpreted as the secondary orders from the complex order ground states. Our work sheds light on the nature of the topological charge density waves in the kagome metals AV3Sb5, and may offer useful indications to the experimentally observed charge orders in the future experiments.