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 $(1+\ensuremath{\lambda})$ 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. 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'' ${\ensuremath{\lambda}}_{\text{eff}}$ inferred from the quasiparticle renormalizations differs from the microscopic coupling characterizing these Hamiltonians $\ensuremath{\lambda}$, and could drastically either overestimate or underestimate 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 line shape 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.

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