Migdal-Eliashberg theory for Superconductivity in Moiré Quantum Materials

Accurately modeling superconductivity in moiré materials requires bridging weak- and strong-coupling regimes where frequency-dependent pairing dynamics are essential. All existing theories that reproduce experimental critical temperatures rely on one of two opposite limits of the Eliashberg framework: the Kohn–Luttinger approximation, which assumes a static interaction but retains full momentum dependence of the pairing function, or the Grabowski–Sham formulation, which averages over momentum while preserving the full frequency dependence that captures retardation effects. Yet neither has been shown to faithfully reproduce the parent Eliashberg equations. Here we solve the complete frequency-dependent Eliashberg equations for microscopic models of twisted bilayer graphene and related moiré systems, incorporating both phonon- and plasmon-mediated kernels together with self-consistent Hartree band renormalization. Comparing the resulting Tc, gap anisotropy, and pairing symmetry against each approximation reveals distinct regimes of validity: Kohn–Luttinger captures sign-changing pairing near van Hove singularities, whereas Grabowski–Sham better describes the strong-coupling limit. Deviations at intermediate coupling expose the limits of perturbative approaches and identify the essential ingredients for predictive superconductivity modeling in flat-band systems.