Magnetic neutron scattering from two-dimensional lattice electrons: The case of ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_4$

It is found that the one-band Hubbard model, in the weak- to intermediate-coupling regime, can account qualitatively for magnetic-neutron-scattering experiments in the paramagnetic phase of ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$ ${\mathrm{Sr}}_{\mathit{x}}$ ${\mathrm{CuO}}_4$ when second-neighbor hopping is included. However, the peak positions, which in two dimensions are determined mostly by the band structure, cannot agree quantitatively with the experimental results when concentration-independent band parameters are used. More importantly, while the energy scale of roughly 150 K seen in the experiments can come from second-neighbor hopping, it arises most naturally if one is very close to a magnetic instability. The proximity to a magnetic instability can be checked experimentally by measuring the relative size of the lattice equivalent of 2${\mathit{k}}_{\mathit{F}}$ anomalies that appear closer to the origin in wave-vector space. Such lattice-2${\mathit{k}}_{\mathit{F}}$ anomalies would allow magnetic neutron scattering to become a spectroscopic tool for the two-dimensional Fermi surface. Finally, exact results are also given for the imaginary part of the Lindhard function on the square lattice.