Anatomy of the Self Energy

TL;DR

This paper explores the representation of Green's functions in field theories without Wick's theorem, introducing a two self energy model that captures complex spectral line shapes and offers testable predictions for photoemission data.

Contribution

It presents a Dyson-like self energy formulation for non-Wick theories and introduces a novel two self energy model with specific spectral predictions.

Findings

A Dyson-like representation of Green's functions is always possible in non-Wick theories.

The two self energy model successfully fits photoemission data in cuprates.

The model predicts skewed spectral line shapes with opposite asymmetries in EDC and MDC analyses.

Abstract

The general problem of representing the Greens function $G(k,z)$ in terms of self energy, in field theories lacking Wick's theorem, is considered. A simple construction shows that a Dyson like representation with a self energy $\Sigma(k,z)$ is always possible, provided we start with a spectral representation for $G(k,z)$ for finite sized systems and take the thermodynamic limit. The self energy itself can then be iteratively expressed in terms of another higher order self energy, capturing the spirit of Mori's formulation. We further discuss alternative and more general forms of $G(k,z)$ that are possible. In particular, a recent theory by the author of extremely correlated Fermi liquids at density "$n$", for Gutzwiller projected non canonical Fermi operators obtains a new form of the Greens function: $$G(k,z)= \frac{\{1- \frac{n}{2}\} + \Psi(k,z)}{z - \hat{E}_k - \Phi(k,z)}, $$ with {\em a pair of self energies} $\Phi(z)$ and $\Psi(z)$. Its relationship with the Dyson form is explored. A simple version of the two self energy model was shown recently to successfully fit several data sets of photoemission line shapes in cuprates. We provide details of the unusual spectral line shapes that arise in this model, with the characteristic skewed shape depending upon a single parameter. The EDC and MDC line shapes are shown to be skewed in opposite directions, and provide a testable prediction of the theory.