  We find that the best RVB state of the spin-$\frac{1}{2}$ Kagome
antiferromagnetic Heisenberg model(spin-$\frac{1}{2}$ KAFH) is described by a
$Z_{2}$ gapped mean field ansatz, which hosts a mean field spinon dispersion
very different from that of the widely studied $U(1)$ Dirac spin liquid state.
However, we find that the physical spin fluctuation spectrum calculated from
the Gutzwiller projected RPA(GRPA) theory above such an RVB state is actually
gapless and is almost identical to that above the $U(1)$ Dirac spin liquid
state. We find that such a peculiar behavior can be attributed to the unique
flat band physics on the Kagome lattice, which makes the mapping between the
mean field ansatz and the RVB state non-injective. We find that the spin
fluctuation 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 the
inelastic neutron scattering(INS) spectrum of Hebertsmithite
ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$, which has been attributed to the contribution of
Cu$^{2+}$ impurity spins occupying the Zn$^{2+}$ site, should rather be
understood as the intrinsic contribution from the Kagome layer. We propose to
verify such a picture by measuring the Knight shift on the Cu site, rather than
the O site, which is almost blind to the spin fluctuation at the $\mathbf{M}$
point as a result of the strong antiferromagnetic correlation between nearest
neighboring spins on the Kagome lattice.
