Momentum-Resolved d-wave Eliashberg Calculation Using The Spin Excitation Spectrum for LSCO Superconductors

We solve the momentum resolved d-wave Elishberg equation employing the magnetic excitation spectrum from the inelastic neutron scattering on the LSCO superconductors reported by Vignolle et al. [1]. The magnetic excitation spectrum exhibits 2 peaks: a sharp incommensurate peak at 18 meV at momentum ($\pi,\pi\pm\delta$) and ($\pi\pm\delta,\pi$), and another broad peak near 40$\sim$70 meV at momentum ($\pi,\pi$). Above 70 meV, the magnetic excitation spectrum has a long tail that is shaped into a circle centered at ($\pi,\pi$) with $\delta '$. The sign of the real part of the total self-energy $\Sigma(\vec{k},\omega)+X(\vec{k},\omega)$ is determined by the momentum position of the peaks of the magnetic excitation spectrum and bare dispersion $\xi(\vec{k})$. We will discuss the effects of the each component of the magnetic excitation spectrum on the self-energy $\Sigma(\vec{k},\omega)$ the renormalization of the band dispersion $X(\vec{k},\omega)$, the pairing function $\phi(\vec{k},\omega)$, and the spectral function $A(\vec{k},\omega)$.\\[4pt] [1] B.Vignolle et.al., Nature Physics 3,163-167 (2007)