Magnetic collective mode dispersion in high-temperature superconductors

Recent neutron-scattering experiments in the superconducting state of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\mathrm{x}}$ have been interpreted in terms of a magnetic collective mode whose dispersion relative to the commensurate wave vector has a curvature opposite in sign to a conventional magnon dispersion. The purpose of this article is to demonstrate that simple linear-response calculations are in support of a collective-mode interpretation, and to explain why the dispersion has the curvature it does.