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

TL;DR

This study explores electron-phonon spectral functions outside the Migdal limit, revealing that effective coupling estimates can be misleading and that perturbation theory remains useful under certain conditions.

Contribution

It provides a comprehensive analysis of spectral functions beyond the Migdal approximation, highlighting discrepancies in coupling measures and validating perturbation theory in specific regimes.

Findings

Effective coupling from quasiparticle renormalizations can significantly misestimate the true coupling.

Perturbation theory accurately predicts spectral features at low coupling and small momenta.

Self-energy and bare electronic structures can be inferred from lineshape analysis and self-consistent fitting.

Abstract

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.