Eliashberg Calculation of the Momentum-Resolved Self-Energy for the Cuprate Superconductors Induced by the Spin Fluctuations

We solve the momentum resolved d-wave Eliashberg equation employing the magnetic excitation spectrum from the inelastic neutron scattering on the LSCO superconductors reported by Vignolle et al. The magnetic excitation spectrum exhibits 2 peaks: a sharp incommensurate peak at 18 meV at momentum (π, π±δ) and (π±δ, π) and another broad peak near 40-70 meV at momentum (π,π). Above 70 meV, the magnetic excitation spectrum has a long tail that is shaped into a circle centered at (π,π) with δ. The sign of the real part of the self-energy is determined by the momentum position of the peaks of the magnetic excitation spectrum and bare dispersion. We will discuss the effects of the each component of the magnetic excitation spectrum on the self-energy, the pairing self-energy.