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This can be easily confirmed by exact diagonalization of the 12-site Heisenberg model on a pinwheel. One finds that the lowest triplet state has an excitation energy of 0 . 260 J . The spectral weight is spread over a large num- ber of states. The highest spectral weight of any one single state is only about 5 percent. The states with the highest 36 spectral weights are spread over the energy range 0 . 260 J < ϵ < 2 . 039 J and they contribute above 80 percent to the spectral weight. If we look at total spec- tral weight up to some energy, roughly 95 . 5 percent of the weight extends up to an energy of 2 . 25 J, 97 . 7 percent of the weight extends up to an energy of 2 . 5 J and roughly 99 . 7 percent of the weight extends up to an energy of 3 J. This strongly confirms that the dominant spectral contributions come from the kink-antikink states. 

In the Valence Bond Crystal state, these triplets can ultimately decay to still lower lying light triplets,[29] though those decay times are likely to be very long, be- cause the pinwheels are surrounded by dimerized trian- gles and have no empty triangles in their immediate vicin- ity, which strongly reduces quantum fluctuations. In the liquid phase, these excitations should have a shorter fi- nite lifetime, which one could represent by a Lorentzian broadening. The frequency dependence of the angle av- eraged dynamic structure factor, for several q-values, at a small Lorentzian broadening (0.01 J) is shown in Fig 3, where as at a larger broadening (0.10 J) it is shown in Fig. 4. The latter may be more representative of the liquid state. One finds that at higher broadening one has a spectral weight that is spread roughly continuously up to an energy of about 3 J . 

For any pair of spins at a distance r , the angle averaged value of exp ( i⃗ q · ⃗ r ) is given by sin ( qr ) /qr . This can be 

used to calculate the angle averaged dynamic structure factor for any given q . For all frequencies, the struc- ture factor as a function of wavenumber shows depen- dence, which is very close to that of an isolated dimer (See Fig. 5). In an isolated pinwheel, the equal-time correlation function is strictly that of a dimer, but ex- citations are extended over the full pin-wheel. Thus the energy integrated structure factor is strictly that of an isolated dimer, but not the spectral weight at a given en- ergy. However, what the calculations show is that while the spectral weight is spread out over a wide range of frequencies, the q-dependence is always near that of an isolated dimer. Since delta chains are likely to be ubiq- uitous in the short-range Valence Bond ordered phase of the Kagome Lattice Heisenberg model,[21] this spectral feature may persist up to energy comparable to J , that is, as long as the system has short range Valence Bond order. 

This picture implies that the equal-time spin-spin cor- relations in the system are essentially only nearest neigh- bor. However, the arrangement of dimerized triangles makes any triplet excitation break into a kink-antikink pairs. These lead to spectral weight spread over a wide frequency range. 

In fact, the spectra obtained by exact-diagonalization of 24, 30 and 36-site clusters by Laeuchli and Lhuillier[19] show very similar frequency dependence, where much of the spectral weight is nearly uniformly spread between J/ 4 to about 2 . 5 J . The finite size system has only short- range Valence bond order. This is further evidence that almost all states with short range Valence Bond order have these features. 

One also knows that the Herbertsmithite materi- als have a small Dzyaloshinski-Moria anisotropy.[30–32] These anisotropies would strongly influence the low en- ergy spectra and the nature of long-range order without significantly altering the high energy spectral properties. In the experiments, the reduction in spectral weight at low temperatures below an energy of 4 meV may well be related to the DM anisotropy which would cause the low energy spectral weight to move to even lower energies. 

Furthermore, the true signature of long-range VBC order, would be the observation of low energy light triplets, which live on the perfect hexagon and bridging dimers.[17, 29] In the VBC phase these provide an ex- tended network for the triplets to move around, whereas the pinwheels form isolated pristine regions which are nearly protected from quantum fluctuations, and have only localized triplets. The honeycomb VBC state has 50 percent of the spectral weight in the non-fluctuating dimer-like heavy excitations and the rest of the 50 per- cent of the spectral weight in the light mobile triplets, which with long-range VBC order have an energy gap of order J/ 20.[14, 15] 

In conclusion, we have argued that the observation of dimer-like q-dependence combined with a spectral weight 

