Approximate calculation of the excitation spectrum of the Heisenberg linear chain

An approximate calculation of the excitation spectrum of the Heisenberg linear chain is performed by the Green's-function technique. Apart from the usual transverse Green's function, a longitudinal Green's function is also found useful. The decoupling procedure is not based on the two-sublattice picture, but utilizes the knowledge of known exact results about the ground state of the linear chain and the insight obtained from a model with explicitly known ground state. The equations are solved in a self-consistent manner, and the spectrum, except near the edge of the Brillouin zone, compares well the exact des Cloizeaux-Pearson result. We also provide an expression for the intensity for neutron-scattering studies; this can be checked against measurements on Cu${\mathrm{Cl}}_2$·2N(${\mathrm{C}}_5$ ${\mathrm{D}}_5$) (CPC).