Diagrammatic analysis of the Hubbard model II: Superconducting state

TL;DR

This paper extends diagrammatic analysis to the superconducting state of the Hubbard model, establishing exact relations between Green's functions, thermodynamic potential, and correlation functions, providing a theoretical framework for understanding superconductivity in this model.

Contribution

It introduces a generalized diagrammatic approach for the superconducting state of the Hubbard model, including new notions of tunneling Green's function and skeleton diagrams, and proves key exact relations.

Findings

Existence of an exact relation between Green's function and thermodynamic potential.

Introduction of skeleton diagrams constructed from irreducible Green's functions.

Proof of stationarity of the thermodynamic potential with respect to the mass operator.

Abstract

Diagrammatic analysis for normal state of Hubbard model proposed in our previous paper [1] is generalized and used to investigate superconducting state of this model. We use the notion of charge quantum number to describe the irreducible Green's function of the superconducting state. As in the previous paper we introduce the notion of tunneling Green's function and of its mass operator. This last quantity turns out to be equal to correlation function of the system. We proved the existence of exact relation between renormalized one-particle propagator and thermodynamic potential which includes integration over auxiliary interaction constant. The notion of skeleton diagrams of propagator and vacuum kinds were introduced. These diagrams are constructed from irreducible Green's functions and tunneling lines. Identity of this functional to the thermodynamic potential has been proved and the stationarity with respect to variation of the mass operator has been demonstrated.