Loop-current charge density wave driven by long-range Coulomb repulsion on the kagomé lattice

Recent experiments on vanadium-based nonmagnetic kagom\'e metals AV₃Sb₅4pt0ex(A=K,Rb,Cs) revealed evidence for possible spontaneous time-reversal symmetry (TRS) breaking in the charge density wave (CDW) ordered state. The long-sought-after quantum order of loop currents has been suggested as a candidate for the TRS breaking state. However, a microscopic model for the emergence of the loop-current CDW due to electronic correlations is still lacking. Here, we calculate the susceptibility of the real and imaginary bond orders on the kagom\'e lattice near van Hove filling, and reveal the importance of next-nearest-neighbor Coulomb repulsion V₂ in triggering the instability toward imaginary bond ordered CDW. The concrete effective single-orbital t-V₁-V₂ model on the kagom\'e lattice is then studied, where t and V₁ are the hopping and Coulomb repulsion on the nearest-neighbor bonds. We obtain the mean-field ground states, analyze their properties, and determine the phase diagram in the plane spanned by V₁ and V₂ at van Hove filling. The region dominated by V₁ is occupied by a 2a₀×2a₀ real CDW insulator with the inverse of Star-of-David (ISD) bond configuration. Increasing V₂ indeed drives a first-order transition from ISD to stabilized loop-current insulators that exhibit four possible current patterns of different topological properties, leading to orbital Chern insulators. We then extend these results away from van Hove filling and show that electron doping helps the stabilization of loop currents, and gives rise to doped orbital Chern insulators with emergent Chern Fermi pockets carrying large Berry curvature and orbital magnetic moment. Our findings provide a concrete model realization of the loop-current Chern metal at the mean-field level for the TRS breaking normal state of the kagom\'e superconductors.