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Spin-spectral-weight distribution and energy range of the parent compound La$_2$CuO$_4$

The spectral-weight distribution in recent neutron scattering experiments on the parent compound La$_2$CuO$_4$ (LCO), which are limited in energy range to about 450\,meV, is studied in the framework of the Hubbard model on the square lattice. We find that the higher-energy weight extends to about 566\,meV and is located at and near the momentum $[π,π]$. Our results confirm that the $U/t$ value suitable to LCO is in the range $U/t\in (6,8)$. The continuum weight energy-integrated intensity vanishes or is extremely small at momentum $[π,0]$. This behavior of the intensity is consistent with that of spin waves, which are damped at $[π,0]$.