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 with effective nearest-neighbor transfer integral $t$ and on-site repulsion $U$. Our study combines a number of numerical and theoretical approaches, including, in addition to standard treatments, density matrix renormalization group calculations for Hubbard cylinders and a suitable spinon approach for the spin excitations. The latter spin-$\frac{1}{2}$ spinons are the spins of the rotated electrons that singly occupy sites. These rotated electrons are mapped from the electrons by a uniquely defined unitary transformation, in which rotated-electron single and double occupancy are good quantum numbers for finite interaction values. Our results confirm that the $U/8t$ magnitude suitable to LCO corresponds to intermediate $U$ values smaller than the bandwidth $8t$, which we estimate to be $8t\approx 2.36$ eV for $U/8t\approx 0.76$. This confirms the unsuitability of the conventional linear spin-wave theory. Our theoretical studies provide evidence for the occurrence of ground-state $d$-wave spinon pairing in the half-filled Hubbard model on the square lattice. This pairing applies only to the rotated-electron spin degrees of freedom, but it could play a role in a possible electron $d$-wave pairing formation upon hole doping. We find that the higher-energy spin spectral weight extends to about 566 meV and is located at and near the momentum $[\pi ,\pi ]$. The continuum weight energy-integrated intensity vanishes or is extremely small at momentum $[\pi ,0]$. This behavior of this intensity is consistent with that of the spin waves observed in recent high-energy neutron scattering experiments, which are damped at the momentum $[\pi ,0]$. We suggest that future LCO neutron scattering experiments scan the energies between 450 and 566 meV and momenta around $[\pi ,\pi ]$.

• Received 27 January 2012

DOI:https://doi.org/10.1103/PhysRevB.86.064520