Variational study of the magnetization plateaus of the spin-$\frac{1}{2}$ kagome Heisenberg antiferromagnet and its implication on YCOB

Numerical simulations find that there are multiple plateaus in the magnetization curve of the spin-$\frac{1}{2}$ Kagome antiferromagnetic Heisenberg model(KAFH) at fractional magnetization $m=1/9,1/3,5/9,7/9$. While it is well known that the $m=1/3,5/9,7/9$ plateau feature a $\sqrt{3}\times\sqrt{3}$ valence bond crystal(VBC) ordering pattern with a David-star-shaped motif, the origin of the narrow plateau at $m=1/9$ remains elusive. Some researchers claim that a subtle translational symmetry breaking pattern with the same $\sqrt{3}\times\sqrt{3}$ periodicity occurs at the $m=1/9$ plateau. On the other hand, it has also been argued that the $m=1/9$ plateau may harbor a novel $Z_{3}$ chiral spin liquid phase. To resolve this controversy, we have proposed the most general variational ansatz based on the resonating valence bond(RVB) picture that is consistent with the spin symmetry of the system and developed a new algorithm to optimize such a complicated ansatz. We find that a peculiar VBC state with a $3\times3$ periodicity and a windmill-shaped motif has significantly lower energy than the claimed $Z_{3}$ chiral spin liquid state and other proposed VBC states around the $m=1/9$ plateau. We find that there are strong spatial modulation in the local magnetization at the $1/9$ plateau, so strong that even its polarization can be reversed. Our general RVB ansatz also well reproduces all other more conventional magnetization plateaus of the spin-$\frac{1}{2}$ KAFH. We find that the local magnetization is always strongly inhomogeneous below the saturating field for such a strongly frustrated quantum magnet.