A spectral function tour of electron-phonon coupling outside the Migdal limit

We simulate spectral functions for electron-phonon coupling in a filled band system - far from the asymptotic limit often assumed where the phonon energy is very small compared to the Fermi energy in a parabolic band and the Migdal theorem predicting 1+lambda quasiparticle renormalizations is valid. These spectral functions are examined over a wide range of parameter space through techniques often used in angle-resolved photoemission spectroscopy (ARPES). Analyzing over 1200 simulations we consider variations of the microscopic coupling strength, phonon energy and dimensionality for two models: a momentum-independent Holstein model, and momentum-dependent coupling to a breathing mode phonon. In this limit we find that any `effective coupling', lambda_eff, inferred from the quasiparticle renormalizations differs from the microscopic dimensionless coupling characterizing these Hamiltonians, lambda, and could drastically either over- or under-estimate it depending on the particular parameters and model. In contrast, we show that perturbation theory retains good predictive power for low coupling and small momenta, and that the momentum-dependence of the self-energy can be revealed via the relationship between velocity renormalization and quasiparticle strength. Additionally we find that (although not strictly valid) it is often possible to infer the self-energy and bare electronic structure through a self-consistent Kramers-Kronig bare-band fitting; and also that through lineshape alone, when Lorentzian, it is possible to reliably extract the shape of the imaginary part of a momentum-dependent self-energy without reference to the bare-band.