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Kagom${\'e}$ lattice is a fertile platform for topological and intertwinedelectronic excitations. Recently, experimental evidence of an unconventionalcharge density wave (CDW) is observed in a Z2 kagom${\'e}$ metalAV$_{3}$Sb$_{5}$ (A= K, Cs, Rb). This observation triggers wide interests onthe interplay between frustrated crystal structure and Fermi surfaceinstabilities. Here we analyze the lattice effect and its impact on CDW inAV$_{3}$Sb$_{5}$. Based on published experimental data, we show that the CDWinduced structural distortions is consistent with the theoretically predictedinverse star-of-David pattern, which preserves the $D_{6h}$ symmetry in thekagom${\'e}$ plane but breaks the sixfold rotational symmetry of the crystaldue to the phase shift between kagom${\'e}$ layers. The coupling between thelattice and electronic degrees of freedom yields a weak first order structuraltransition without continuous change of lattice dynamics. Our result emphasizesthe fundamental role of lattice geometry in proper understanding ofunconventional electronic orders in AV$_{3}$Sb$_{5}$.

Geometry of the charge density wave in kagom${\'e}$ metal AV$_{3}$Sb$_{5}$