A generating functional approach to the Hubbard model

TL;DR

This paper extends the generating functional method to strongly correlated systems, specifically applying it to the Hubbard model to analyze Green's functions, collective modes, and phase instabilities.

Contribution

It develops a generalized generating functional approach for the Hubbard model, deriving equations for Green's functions and collective modes, including corrections and mean field approximations.

Findings

Derived equations for electronic and Bose-like Green's functions.

Calculated self-energy corrections up to second order in W/U.

Identified a soft mode indicating potential charge ordering instability at half-filling.

Abstract

The method of generating functional is generalized to the case of strongly correlated systems, and applied to the Hubbard model. For the electronic Green's function constructed for Hubbard operators, an equation using variational derivatives with respect to the fluctuating fields has been derived and its multiplicative form has been determined. Corrections for the electronic self-energy are calculated up to the second order with respect to the parameter W/U (W width of the band), and a mean field type approximation was formulated, including both charge and spin static fluctuations. The equations for the Bose-like Green's functions have been derived, describing the collective modes: the magnons and doublons. The properties of the poles of the doublon Green's functions depend on electronic filling. The investigation of the special case n=1 demonstrates that the doublon Green's function has a soft mode at the wave vector Q=(pi,pi,...), indicating possible instability of the uniform paramagnetic phase relatively to the two sublattices charge ordering. However this instability should compete with an instability to antiferromagnetic ordering.