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We find that the best RVB state of the spin-$\frac{1}{2}$ Kagomeantiferromagnetic Heisenberg model(spin-$\frac{1}{2}$ KAFH) is described by a$Z_{2}$ gapped mean field ansatz, which hosts a mean field spinon dispersionvery different from that of the widely studied $U(1)$ Dirac spin liquid state.However, we find that the physical spin fluctuation spectrum calculated fromthe Gutzwiller projected RPA(GRPA) theory above such an RVB state is actuallygapless and is almost identical to that above the $U(1)$ Dirac spin liquidstate. We find that such a peculiar behavior can be attributed to the uniqueflat band physics on the Kagome lattice, which makes the mapping between themean field ansatz and the RVB state non-injective. We find that the spinfluctuation spectrum of the spin-$\frac{1}{2}$ KAFH is not at all featureless,but is characterized by a prominent spectral peak at about $0.25J$ around the$\mathbf{M}$ point, which is immersed in a broad continuum extending to $2.7J$.Based on these results, we argue that the spectral peak below 2 meV in theinelastic neutron scattering(INS) spectrum of HebertsmithiteZnCu$_{3}$(OH)$_{6}$Cl$_{2}$, which has been attributed to the contribution ofCu$^{2+}$ impurity spins occupying the Zn$^{2+}$ site, should rather beunderstood as the intrinsic contribution from the Kagome layer. We propose toverify such a picture by measuring the Knight shift on the Cu site, rather thanthe O site, which is almost blind to the spin fluctuation at the $\mathbf{M}$point as a result of the strong antiferromagnetic correlation between nearestneighboring spins on the Kagome lattice.

physical review. b./physical review. b

Variational study of the ground state and spin dynamics of the  spin-$\frac{1}{2}$ Kagome antiferromagnetic Heisenberg model and its  implication on Hebertsmithite ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$