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 $m=1/3,5/9,7/9$ plateaus 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. However, it has also been argued that the $m=1/9$ plateau may harbor a novel $Z_3$ chiral spin-liquid phase. The main challenge to resolve this controversy is to reduce the bias of the method used to describe the magnetization process of such a strongly frustrated quantum magnet, for which intricate competition between different symmetry-breaking patterns is generally expected. For this purpose, 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. To optimize such a general RVB ansatz, which can describe all kinds of spatial symmetry-breaking pattern on equal footing but at the same time contains a huge number of variational parameters, we have developed a new optimization algorithm, the finite-depth Broyden-Fletcher-Goldfarb-Shanno algorithm. The new algorithm can achieve good balance between numerical efficiency, numerical stability, and storage demand and can deal with optimization problem containing millions of parameters at ease. We applied such an advanced optimization algorithm to the general RVB ansatz and mapped out the magnetization curve of the spin-$\frac{1}{2}$ KAFH. We find that a peculiar VBC state with a $3\times 3$ 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 modulations 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.

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• Antiferromagnetism
• Frustrated magnetism
• Magnetism
• Quantum spin liquid
• Antiferromagnets
• Strongly correlated systems
• Lattice models in condensed matter
• Monte Carlo methods
• Trial wave functions
• Variational wave functional methods