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Understanding the nature of charge density wave (CDW) and superconductivityin kagome metal AV$_3$Sb$_5$ (A=Cs,Rb,K) is a recent subject of intensivestudy. Due to the presence of van Hove singularities, electron-electroninteraction has been suggested to play an important role in the formation ofsuch broken symmetry states. Recent experiments show that the CDW order isthree-dimensional and it is staggered across different kagome layers. However,the experimental interpretation for the precise structure of CDW varies interms of whether it is the star of David (SD), inverse star of David (ISD) orthe alternation of the two among neighboring layers. In this work, we show thatthe origin of these distinct CDW orders can be understood in a unified pictureby considering intra- and inter-layer electron-electron interactions as well asthe coupling between electrons and lattice distortions. Utilizing an effective9-band model with V $d$ orbitals and out-of-plane Sb $p$ orbitals, it isdemonstrated that the repulsive electron-electron interaction favors chargebond order which induces either SD or ISD upon including lattice distortions.As the inter-layer interaction is introduced, $\pi$-shifted CDW develops withthe staggered ordering along the $c$-axis. We also find that the phase withalternating SD and ISD can be stabilized as the ground state under stronginter-layer interaction.

Origin of $\pi$-shifted three-dimensional charge density waves in kagome metal AV$_3$Sb$_5$