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Experiments in the recently discovered vanadium-based kagome metals havesuggested that their charge-ordered state displays not only bond distortions,characteristic of a ``real" charge density-wave (rCDW), but also time-reversalsymmetry-breaking, typical of loop currents described by an ``imaginary" chargedensity-wave (iCDW). Here, we combine density-functional theory, group-theory,and phenomenological modeling to investigate the complex charge-ordered statesthat arise from interactions between the low-energy van Hove singularitiespresent in the electronic structure of $A$V$_3$Sb$_5$. We find two broadclasses of mixed iCDW-rCDW configurations: triple-$\mathbf{Q}$ iCDW,triple-$\mathbf{Q}$ rCDW order, dubbed $3\mathbf{Q}$-$3\mathbf{Q}$, anddouble-$\mathbf{Q}$ iCDW, single-$\mathbf{Q}$ rCDW order, dubbed$2\mathbf{Q}$-$1\mathbf{Q}$. Moreover, we identify seven different types ofiCDW order, stemming from the different vanadium-orbital and kagome-sublatticestructures of the two pairs of van Hove singularities present above and belowthe Fermi level. While the $2\mathbf{Q}$-$1\mathbf{Q}$ states trigger anorthorhombic distortion that breaks the threefold rotational symmetry of thekagome lattice, the $3\mathbf{Q}$-$3\mathbf{Q}$ states induce various types ofsubsidiary uniform magnetic orders, from conventional ferromagnetism tomagnetic octupolar, magnetic toroidal, and even magnetic monopolar order. Weshow that these exotic orders display unique magneto-striction,magneto-electric, and magneto-electric-striction properties that can be probedexperimentally to identify which iCDW state is realized in these compounds. Webriefly discuss the impact of an out-of-plane modulation of the charge orderand the interplay between these complex charge-ordered states andsuperconductivity.

Loop currents in $A$V$_3$Sb$_5$ kagome metals: multipolar and toroidal magnetic orders