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Vanadium-based materials AV$_3$Sb$_5$ (A=K, Rb, Cs) with layered kagomelattice structures have drawn great attention recently due to the discoveriesof topologically nontrivial band structures, charge density wave states, giantanomalous Hall effect, as well as unusual superconducting phase at lowtemperatures. In this work, we theoretically study the magnetic and chargeinstabilities for this class of materials based on first principlescalculations. We develop a method to calculate the generalized susceptibilitytensor defined in the sublattice-orbital-spin space, with the effects ofCoulomb interactions treated by generalized random phase approximation (RPA).The RPA susceptibility calculations indicate that there are three leadingferromagnetic instability modes at $\Gamma$ point, which are further verifiedby unrestricted self-consistent Hartree-Fock calculations including both theon-site and inter-site Coulomb interactions. The inclusion of inter-siteinteractions tend to suppress the spin ferromagnetism due to charge transferfrom the V to Sb sites, leading to weak spin magnetic moments $\sim0.1\mu_{\textrm{B}}$ per V atom with small intrinsic anomalous Hallconductivity. Current loops can be generated in such weak spin ferromagneticstates as a result of spin-orbit coupling effects. The electronic structures inthe ferromagnetic states are significantly reconstructed which have nearlycompensated electron and hole carriers from two bands. On the other hand, we donot find any diverging instability mode at $M$ point driven byelectron-electron Coulomb interactions. First principles phonon calculationsindicate that there are unstable phonon modes which tend to drive the systeminto an inverse star-of-David structure. Our results indicate that there may beseparate phase transitions in the magnetic and charge channels in the system.

First principles theory for the magnetic and charge instabilities in AV$_3$Sb$_5$ systems