https://doi.org/10.1140/epjb/e2005-00166-7
A generating functional approach to the Hubbard model
1
Institute for Metal Physics, Ural Division of the Russian
Academy of Sciences, 620219, Ekaterinburg, S. Kovalevskaya
str.18, Russia
2
Dipartimento di Fisica “E.R. Caianiello", Università
degli Studi di Salerno, Via S. Allende, 84081 Baronissi (SA),
Italy
Corresponding author: a mancini@sa.infn.it
Received:
22
November
2004
Revised:
17
February
2005
Published online:
16
June
2005
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
, 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.
PACS: 71.10.-w – Theories and models of many-electron systems / 71.10.Fd – Lattice fermion models (Hubbard model, etc.) / 71.27.+a – Strongly correlated electron systems; heavy fermions
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2005