A generating functional approach to the Hubbard model

The method of generating functional, suggested for conventional systems by Kadanoff and Baym, is generalized to the case of strongly correlated systems, described by the Hubbard X operators. The method has been applied to the Hubbard model with arbitrary value U of the Coulomb on-site interaction. For the electronic Green’s function constructed for Fermi-like X operators, an equation using variational derivatives with respect to the fluctuating fields has been derived and its multiplicative form has been determined. The Green’s function is characterized by two quantities: the self energy Σ and the terminal part Λ. For them we have derived the equation using variational derivatives, whose iterations generate the perturbation theory near the atomic limit. Corrections for the electronic self-energy Σ 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. This approximation is actually equivalent to the one used in the method of Composite Operators, and it describes an insulator-metal phase transition at half filling reasonably well. The equations for the Bose-like Green’s functions have been derived, describing the collective modes: the magnons and doublons. The main term in this equation represents variational derivatives of the electronic Green’s function with respect to the corresponding fluctuating fields. 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=(π,π,...), 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. The generating functional method with the X operators could be extended to the other models of strongly correlated systems.