The Zubarev Double Time Greens function-A Vintage Many Body Technique

TL;DR

This work presents a pedagogical development of Zubarev's Double Time Green's Function method for quantum many-body problems, outlining definitions, spectral representations, and the connection between Green's functions, correlation functions, and spectral densities. It demonstrates the formalism on canonical models, including the non-interacting electron and phonon gases, and discusses how the Hubbard model reveals the Stoner criterion, with explicit links to self-energies and observable distributions. The text emphasizes the inverse-operator view of Green's functions, the handling of higher-order hierarchies via truncation, and the extendability of the approach to superconductivity and other correlated phenomena. Together, these notes provide a coherent, versatile toolkit for analyzing interacting fermionic and bosonic systems within a DTGF framework and underlie foundational results in condensed matter theory.

Abstract

These lecture notes present a comprehensive and powerful many-body technique pioneered in 1960 by D. N. Zubarev. The technique, known as the Zubarev Double Time Greens Function method, was used extensively by leading solid state physicists such as John Hubbard and Laura Roth in the 1960s. We present the technique and apply it to the non-interacting electron and boson gas. We next consider the (many-body) Hubbard model and show how it yields the Stoner criterion for ferromagnetism. It is easily extendable to superconductivity and related problems. Our treatment is pedagogical and understandable to those with just an elementary understanding of second quantization.

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