Loop currents in AV₃Sb₅ kagome metals: Multipolar and toroidal magnetic orders

Experiments in the recently discovered vanadium-based kagome metals have suggested that their charge -ordered state displays not only bond distortions, characteristic of a "real" charge density wave (rCDW), but also time-reversal symmetry breaking, typical of loop currents described by an "imaginary" charge density wave (iCDW). Here, we combine density-functional theory, group theory, and phenomenological modeling to investigate the complex charge-ordered states that arise from interactions between the low-energy van Hove singularities present in the electronic structure of AV3Sb5. We find two broad classes of mixed iCDW-rCDW configurations: triple -Q iCDW, triple -Q rCDW order, dubbed 3Q-3Q, and double -Q iCDW, single -Q rCDW order, dubbed 2Q-1Q. Moreover, we identify seven different types of iCDW order, stemming from the different vanadium-orbital and kagome-sublattice structures of the two pairs of van Hove singularities present above and below the Fermi level. While the 2Q-1Q states trigger an orthorhombic distortion that breaks the threefold rotational symmetry of the kagome lattice, the 3Q-3Q states induce various types of subsidiary uniform magnetic orders, from conventional ferromagnetism to magnetic octupolar, magnetic toroidal, and even magnetic monopolar order. We show that these exotic orders display unique magnetostriction, magnetoelectric, and magnetoelectrostriction properties that can be probed experimentally to identify which iCDW state is realized in these compounds. We briefly discuss the impact of an out-of-plane modulation of the charge order and the interplay between these complex charge-ordered states and superconductivity.