Structure Factors of the Kagome-Lattice Heisenberg antiferromagnets at finite temperatures.

We compute the real-space spin correlations and frquency and wave-vector resolved dynamic structure factors $S(\vec q,\omega)$ for the nearest-neighbor Kagome-Lattice Heisenberg Antiferromagnet (KLHM) at finite temperatures using Numerical Linked Cluster Expansion (NLCE) method. A triangle-based NLCE is used to calculate frequency moments of the dynamic structure factors in the thermodynamic limit, which show excellent convergence for $T> J/4$. A Gaussian approximation and the fluctuation-dissipation relation are used to to reconstruct the frequency dependence. We find that some features of the low temperature KLHM structure-factors begin to set in at temperatures of order $J$. Our results are in very good agreement with powder diffraction measurements reported earlier on the Herbertsmithite materials ZnCu$_3$(OH)$_6$Cl$_2$. However, the calculated properties differ from the low temperature ($T\approx J/100$) experimental measurements in one important regard. In line with the experimental observations, the spectral weight has a diffuse nature, which is predominantly spread along the extended Brillouin-Zone boundary. However, the maximum intensity is found in our calculations to be at the $K$ point of the extended Brillouin Zone in contrast to the low temperature experiments, where it is at the $M$ point. We suggest that experiments should be done at various temperatures to look for such a crossover of the maximum from the $K$ point to the $M$ point. In the absence of such a crossover, the Herbertsmithite materials must differ from KLHM in a significant manner.