Anatomy of the self-energy

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 $\ensuremath{\Sigma}(k,z)$ 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 noncanonical Fermi operators, obtains a new form of the Greens function: $G(k,z)=[(1\ensuremath{-}\frac{n}{2})+\ensuremath{\Psi}(k,z)]/[z\ensuremath{-}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{k}\ensuremath{-}\ensuremath{\Phi}(k,z)],$ with a pair of self-energies $\ensuremath{\Phi}(z)$ and $\ensuremath{\Psi}(z)$. Its relationship with the Dyson form is explored. A simple version of the two-self-energies 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 energy distribution curve (EDC) and momentum distribution curve (MDC) line shapes are shown to be skewed in opposite directions, and provide a testable prediction of the theory.