LAUSR

We present a novel method for embedding spin and charge fluctuations in an anisotropic, multiband, and full-bandwidth Eliashberg treatment of superconductivity. Our analytical framework, based on the random phase approximation, allows for a self-consistent calculation of material-specific characteristics in the interacting, and more specifically, the superconducting state. We apply this approach to bulk FeSe as a representative for the iron-based superconductors and successfully solve for the superconducting transition temperature ${T}_{c}$, the gap symmetry, and the gap magnitude. We obtain ${T}_{c}\ensuremath{\approx}6$ K, consistent with experiment (${T}_{c}\ensuremath{\approx}8$ K), as well as other quantities in good agreement with experimental observations, thus supporting spin fluctuations mediated pairing in bulk FeSe. On the contrary, applying our approach to monolayer FeSe on ${\mathrm{SrTiO}}_{3}$ we find that spin fluctuations within the full Eliashberg framework give a $d$-wave gap with ${T}_{c}\ensuremath{\le}11$ K and therefore cannot provide an explanation for a critical temperature as high as observed experimentally (${T}_{c}\ensuremath{\approx}70$ K). Our results hence point towards interfacial electron-phonon coupling as the dominant Cooper pairing mediator in this system.